Rule: The contr from a pure curve Ga of 1/r(1,r-1) is

1/Denom([1,1,r,r])
 * ( -b*t^(r+1) + N*t^3*((1-t^(r-2)) div (1-t)) * ((1-t^r) div (1-t)) );

where b = deg Ga (in units of 1/r^2) and deg_Ga(E1-E{r-1}) = 2*r*N - 2*b.

Or
  -b*t^(r+1)/11rr
  +N*t^3*(1-t^(r-2))/1111r
where b = deg Ga (in units of 1/r^2) and N = 1/r(deg_Ga(E1-E{r-1})/2 + b).
The first term is exactly b*Qorb. The second term does not depend on b.

Check: 2g(Ga)-2 = deg_Ga(-E1-E{r-1})

Notes: This is valid for r = 2, but then it doesn't involve N, and
simplifies to

 b* -t^3 / &*[1-t^i : i in [1,1,2,2]];

If E1-E{r-1} = Oh_Ga(al) then the term deg_Ga(E1-E{r-1}) is al * b 
and
 N = (al+2)*b / 2*r;
this is a problem in specifying N, b as integers: we have to take al and
b so that (al + 2)*B is divisible by 2*r.
The genus of Ga is present here: by adjunction
  K_Ga = K_Y + E1 + E{r-1}, so (for CY) 2g-2 = deg_Ga(E1+E{r-1}) = 2g-2.

Magma:

/* Contr of a 1/r(1,r-1) curve Ga is
N * t^3 * ((1-t^(r-2)) div (1-t)) * ((1-t^r) div (1-t))
- b * t^{r+1}
all over Denom([1,1,r,r]). Note the funny way of putting in the
parameters: b = deg Ga and
 deg_Ga(E1-E{r-1}) = 2*r*N - 2*b.  */

function Ppc(r,N,b)
X := t^3*((1-t^(r-2)) div (1-t)) * ((1-t^r) div (1-t));
Y := -t^(r+1);
return (N*X+b*Y)/Denom([1,1,r,r]);
end function;

===============

There are 5 pure curves 1/6(1,5) among the hypersurfaces:
   [1,5,6,6,13,31],
   [1,5,6,6,17,35],
   [1,5,6,6,18,36],
   [5,6,6,7,17,41],
   [5,6,7,12,12,42]

[ 1, 5, 6, 6, 13, 31 ], so b = 1,
(2*t^5 + 2*t^3)
deg E1,E5 = 17,-5, so N = 1/6(11+1) = 2
2g-2 = -(17-5)/6

[ 1, 5, 6, 6, 17, 35 ], so b = 1,
(-t^5 - t^3)
deg E1,E5 = -1,13, so N = 1/6(-7+1) = -1

[ 1, 5, 6, 6, 18, 36 ], so b = 2,
(t^3+t^5)
deg E1,E5 = -1,-5, so N = 1/6(2*2+2) = 1
2g-2 = 2(1+5)/6 = 2

[ 5, 6, 6, 7, 17, 41 ], so b = 1
-2(t^3+t^5)
deg E1,E5 = -7,19, so N = 1/6((-7-19)/2+1) = -2

===============

Rule: curve of 1/5(2,3) contr (N*x + b*y)/[1,1,5,5] with
x := -t^3*((1-t^5) div (1-t))*(3-2*t+3*t^2);
y := t^3*(1+t^2)*(1+t^2+t^4);

where deg Ga = b and deg_Ga (E2-E3) = 10*N - 4*b;

Note: Qorb(5,[2,3],5)/(1-t^5) is (t^7 - t^6 + t^5)/[1,1,5,5]

Alternative Rule: curve of 1/5(2,3) contr
  b*Qorb(5,[2,3],5)/(1-t^5)
  +(N*(-3+2*t-3*t^2) + b*(1-t+t^2))*t^3/[1,1,1,5]

where b = deg Ga and N = 1/5*(deg_Ga(E2-E3)/2+2*b)
NB. 2g-2(C) = deg(E2+E3)/5 so E2-E3 is even

Pure curve of 1/5(2,3) (there are 19 of them among the h'surfaces)
   [2,2,3,5,5,17],
   [2,3,3,5,5,18],
   [2,3,5,5,5,20],
   [2,3,5,5,12,27],
   [2,3,5,5,13,28],
   [2,3,5,5,15,30],
   [2,5,5,8,12,32],
   [3,5,5,7,13,33],
   [3,5,7,10,10,35],
   [3,5,5,7,20,40],
   [2,5,5,8,20,40],
   [3,7,10,10,15,45],
   [5,5,8,12,18,48],
   [2,5,8,10,25,50],
   [5,8,12,15,20,60],
   [5,5,8,12,30,60],
   [5,7,10,13,35,70],
   [5,10,12,18,45,90],
   [5,5,17,18,45,90]

 PC - b*Q5 has denom [1,1,1,5]

[ 2, 3, 3, 5, 5, 18 ] -> b = 1,
(4*t^5 - 3*t^4 + 4*t^3)/[1,1,1,5]
deg E2-E3 is -2,+12, so N = 1/5*(deg(E2-E3)/2+2*b) = -1

[ 2, 3, 5, 5, 5, 20 ] -> b = 4,
(-2*t^5 - 2*t^3)/[1,1,1,5]
deg E2-E3 is -2,-3, so N = 1/5*(4*(-2+3)/2+8) = 2

[ 2, 3, 5, 5, 15, 30 ] -> b = 2,
(-t^5 - t^3)/[1,1,1,5]
deg E2-E3 is -2,-3

[ 2, 5, 5, 8, 20, 40 ] -> b = 2
(-4*t^5 + 2*t^4 - 4*t^3)/[1,1,1,5]
deg E2-E3 is -2,-8, N = 2

[ 2, 5, 8, 10, 25, 50 ] -> b = 1
(-2*t^5 + t^4 - 2*t^3)/[1,1,1,5]
deg E2,E3 is -2,-8

[ 3, 5, 5, 7, 13, 33 ] -> b = 1
(7*t^5 - 5*t^4 + 7*t^3)/[1,1,1,5]
deg E2,E3 is -7,+17

[ 5, 5, 8, 12, 18, 48 ] => b=1
(10*t^5 - 7*t^4 + 10*t^3)/[1,1,1,5]
deg E2,E3 is -12,22

[ 5, 5, 17, 18, 45, 90 ] => b=1
(-t^5 - t^3)/[1,1,1,5]
deg E2,E3 is -17,-18

[ 5, 7, 10, 13, 35, 70 ] => b=1
(-2*t^5 + t^4 - 2*t^3)/[1,1,1,5]
deg E2,E3 is -7,-13

[ 5, 10, 12, 18, 45, 90 ] => b=1
(-2*t^5 + t^4 - 2*t^3)/[1,1,1,5]
deg E2,E3 is -12,-18

[ 2, 3, 5, 5, 12, 27 ] => b=1
(-5*t^5 + 3*t^4 - 5*t^3)/[1,1,1,5]
deg E2,E3 is 13,-3

[ 2, 5, 5, 8, 12, 32 ]
(-8*t^5 + 5*t^4 - 8*t^3)/[1,1,1,5]
deg E2,E3 is 18,-8

[ 3, 5, 5, 7, 20, 40 ] => b=2
(2*t^5 - 2*t^4 + 2*t^3)/[1,1,1,5]
deg E2,E3 is -7,-3

[ 5, 5, 8, 12, 30, 60 ] => b=2
(2*t^5 - 2*t^4 + 2*t^3)/[1,1,1,5]
deg E2,E3 is -12,-8

[ 5, 8, 12, 15, 20, 60 ] => b=1
(t^5 - t^4 + t^3)/[1,1,1,5]
deg E2,E3 is -12,-8

============

Rule: curve of 1/7(2,5) contr (N*x+b*y)/[1,1,7,7] with
x := -t^3*((1-t^7) div (1-t))*(5-4*t+7*t^2-4*t^3+5*t^4);
y := t^3*(1+t^2+t^4)*(1+t^2+t^4+t^6);

where deg Ga = b and deg_Ga(E2-E5) = 14*N - 4*b?

Alternative rule: pure curve of 1/7(2,5) contr
  b*Qorb(7,[2,5],7)/(1-t^7)
 +N*t^3*(-5 + 4*t - 7*t^2 + 4*t^3 - 5*t^4)
 +b*t^3*(1 - t + 2*t^2 - t^3 + t^4)/1,1,1,7

where b = deg Ga (in units of 1/7^2) and N = 1/7(deg(E2-E5)/2+2b)

There are 4 pure 1/7(2,5) curves among the hypersurfaces
   [2,5,7,7,16,37],
   [2,5,7,7,19,40],
   [2,5,7,7,21,42],
   [5,7,9,14,35,70]

[ 2, 5, 7, 7, 19, 40 ] => b=1
(6*t^7 - 5*t^6 + 9*t^5 - 5*t^4 + 6*t^3)/1117
deg E2,E5 is -2,16, N = -1, b = 1.
genus Ga = 0, -2 = (-16+2)/14

[ 2, 5, 7, 7, 21, 42 ] => b = 2;
(-3*t^7 + 2*t^6 - 3*t^5 + 2*t^4 - 3*t^3)
deg E2,E5 is -2,-5, N = 1/7(2*(-2+5)/2+2*2) = 1
genus Ga = 2, 2 = 2(2+5)/7

[ 5, 7, 9, 14, 35, 70 ] => b = 1;
(t^7 - t^6 + 2*t^5 - t^4 + t^3)
deg E2,E5 is -9,-5, so N = 1/7(-9+5)/2+2) = 0
genus Ga = 2 = 1(9+5)/2

============

There are 4 pure 1/7(3,4) curves among the qsm hypersurfaces
   [3,3,4,7,7,24],
   [3,4,7,7,7,28],
   [3,4,7,7,18,39],
   [4,7,10,14,35,70]

[ 3, 3, 4, 7, 7, 24 ], b=1,
(-6*t^7 - 5*t^6 + 3*t^5 - 5*t^4 - 6*t^3) // after taking of the Qorb
deg E3,E4 is 18,-4. So N = 2

[ 3, 4, 7, 7, 7, 28 ], b=4,
(-2*t^6 - 2*t^4)
deg E3,E4 is -3,-4. So N = 2

[ 3, 4, 7, 7, 18, 39 ], b=1
(6*t^7 + 4*t^6 - 3*t^5 + 4*t^4 + 6*t^3)
deg E3,E4 is -3,17. So N = -1

[ 4, 7, 10, 14, 35, 70 ], b=1
(2*t^7 + t^6 - t^5 + t^4 + 2*t^3)
deg E3,E4 is -10,-4. So N = 0

Guess: N is (deg_Ga(E3-E4)/2+3b)/7
and the rule is
  b*(2*t^7 + t^6 - t^5 + t^4 + 2*t^3)
  +N*(-4*t^7 - 3*t^6 + 2*t^5 - 3*t^4 - 4*t^3)

============

For 1/8(3,5) the guess is
  N is deg_Ga(E3-E5)/2+3b)/8
and rule
 b*(-3*t^8 + t^7 - t^6 - t^5 + t^4 - 3*t^3)
 +N*(8*t^7 - 12*t^6 + 17*t^5 - 12*t^4 + 8*t^3)*(1+t)

There are 3 of them
   [3,5,8,8,19,43], b=1, E3,E5 = 21,-5, N = 2
   [3,5,8,8,21,45], b=1, E3,E5 = -3,19, N = -1
   [3,5,8,8,24,48], b=2, E3,E5 = -3,-5 (x b), N = 1

(13*t^7 - 20*t^6 + 29*t^5 - 20*t^4 + 13*t^3)*(1+t)
(-11*t^7 + 16*t^6 - 22*t^5 + 16*t^4 - 11*t^3)*(1+t)
(2*t^7 - 4*t^6 + 7*t^5 - 4*t^4 + 2*t^3)*(1+t)

confirms the guess.

============

There are 5 pure 1/7(1,6) curves among the hypersurfaces
[ 1, 6, 7, 7, 15, 36 ], b=1
(2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3)
deg E1,E6 is 20,-6

[ 1, 6, 7, 7, 21, 42 ], b=2
(t^7 + t^6 + t^5 + t^4 + t^3)
deg E1,E6 = -1,-6

[ 6, 7, 7, 8, 20, 48 ], b=1
(-2*t^7 - 2*t^6 - 2*t^5 - 2*t^4 - 2*t^3)
deg E1,E6 = -8,22

[ 6, 7, 7, 8, 28, 56 ], b=2
0
deg E1,E6 = -8,-6

[ 7, 7, 9, 13, 36, 72 ], b=1
(-6*t^7 + 3*t^6 - 6*t^5 + 3*t^4 - 6*t^3)
deg E1,E6 = -36,-13,


[2,5,7,7,19,40],
d:=40; A:=[2,5,7,7,19]; B:=[[19,5,7,7]]; N:=-1; b:=1;
D := P(d,A) - PA(A) - Pb(B) - (N*X+b*Y)/denom;
 // 6*t^13+t^12+10*t^11+4*t^10+12*t^9+9*t^8+12*t^7+4*t^6+10*t^5+t^4+6*t^3
 // -X+Y; deg Ga = 1; E2 = Oh(-2); E5 = Oh(40-5-19) = Oh(16), E2-E5 = Oh(-18)
 
[2,5,7,7,19,40],
d:=40; A:=[2,5,7,7,19]; B:=[[19,5,7,7]];
D := P(d,A) - PA(A) - Pb(B);
 // 6*t^13+t^12+10*t^11+4*t^10+12*t^9+9*t^8+12*t^7+4*t^6+10*t^5+t^4+6*t^3
 // -X+Y; deg Ga = 1; E2 = Oh(-2); E5 = Oh(40-5-19) = Oh(16), E2-E5 = Oh(-18)

[2,5,7,7,21,42],
d:=42; A:=[2,5,7,7,21]; B:=[[5,1,2,2]];
D := P(d,A) - PA(A) - Pb(B);
 // -3*t^13-t^12-4*t^11-4*t^10-3*t^9-9*t^8-3*t^7-4*t^6-4*t^5-t^4-3*t^3
 // deg Ga = 2; E2 = Oh(-2), E5 = Oh(-5)

[2,5,7,7,21,42],
d:=42; A:=[2,5,7,7,21]; B:=[[5,1,2,2]]; N:=1; b:=2;
D := P(d,A) - PA(A) - Pb(B) - (N*X+b*Y)/denom;
 // -3*t^13-t^12-4*t^11-4*t^10-3*t^9-9*t^8-3*t^7-4*t^6-4*t^5-t^4-3*t^3
 // deg Ga = 2; E2 = Oh(-2), E5 = Oh(-5)

[3,4,7,7,7,28],
d:=28; A:=[3,4,7,7,7]; B := [[3,1,1,1]];
D := P(d,A) - PA(A) - Pb(B);
 // -2*t^12-2*t^11-8*t^8-2*t^5-2*t^4

[4,7,10,14,35,70],
d:=70; A:=[4,7,10,14,35]; B:=[[5,2,4,4]];
D := P(d,A) - PA(A) - Pb(B) - 70/(4*10*14)*Phalfc() + Pemb2(2);


[3,4,7,7,21,42],
d:=42; A:=[3,4,7,7,21]; B:=[[4,
D := P(d,A) - PA(A) - Pb(B);
 // does not have a tangent monomial at y

[3,4,7,7,18,39], // curve of 1/2 with emb 1/4(3,3,2) and 1/18(4,7,7)


[1,4,7,7,9,28], // doesn't actually have a curve of 1/7
d:=28; A:=[1,4,7,7,9]; B := [[9,4,7,7]];
D := P(d,A) - PA(A) - Pb(B);
  // if we pretend the denom is [1,1,7,7] then
  // 4*t^12 + 4*t^10 + 4*t^9 + 4*t^8 + 4*t^7 + 4*t^6 + 4*t^4

[1,10,11,33,55,110],
d := 110; A:=[1,10,11,33,55]; // emb 1/33(1,10,22)

[2,9,11,33,44,99], // emb 1/44
[2,9,11,33,53,108], // emb 1/33
[2,9,11,33,55,110], // emb 1/33(2,9,22)



[2,5,7,14,14,42], // curve of 1/7 with emb 3 x 1/14(2,5,7)
[2,5,7,14,26,54], // curve of 1/7 with emb 1 x 1/14(2,5,7) but also 1/26(5,7,14)
[2,5,14,14,35,70], // curve of 1/7 with emb 5 x 14/(2,5,7)


[5,5,8,12,18,48],
[5,5,17,18,45,90], // curve of 1/5(2,3) pure
d:=90; A:=[5,5,17,18,45]; B:=[[17,1,5,11],[9,5,5,8]];
D := P(d,A) - PA(A) - Pb(B);
 // -t^9 - t^8 - 4*t^6 - t^4 - t^3 = -X + Y
 // deg Ga = 2, E2 = Oh(-17), E3 = Oh(-18), E2-E3 = Oh(1), its deg = 2
 // N = 1; b := 2;

[2,3,3,5,5,18],
d := 18; A :=[2,3,3,5,5]; B:= [[3,2,2,2] : i in [1..6]];
D := P(d,A) - PA(A) - Pb(B);
 // 4*t^9 + t^8 + 6*t^7 + 4*t^6 + 6*t^5 + t^4 + 4*t^3 = 4X - Y
 // deg Ga = 1, E2 = Oh(-2), E3 = Oh(12), E2-E3 = Oh(-14), its deg = -14
 // b := 1, N := -1;

[3,5,5,7,13,33],
d := 33; A:=[3,5,5,7,13]; B:=[[7,3,5,6],[13,3,5,5]];
D := P(d,A) - PA(A) - Pb(B);
 // 7*t^9 + 2*t^8 + 10*t^7 + 8*t^6 + 10*t^5 + 2*t^4 + 7*t^3
 // deg Ga = 1, E2 = Oh(-7), E3 = Oh(33-3-13) = Oh(17), E2-E3 = Oh(24), deg 24

[7,7,9,13,36,72],
d:=72; A:= [7,7,9,13,36]; B:=[[9,7,7,4],[9,7,7,4],[13,7,9,10]];
D := P(d,A) - PA(A) - Pb(B);
 // -6*t^13 - 3*t^12 - 9*t^11 - 6*t^10 - 12*t^9 - 13*t^8
 //   - 12*t^7 - 6*t^6 - 9*t^5 - 3*t^4 - 6*t^3

These check out Ppc

[1,6,7,7,15,36],
d:=36; A:=[1,6,7,7,15]; B:=[[15,1,7,7],[3,1,1,1]];
D := P(d,A) - PA(A) - Pb(B) - Ppc(7,2,1);
 // 2*t^13+4*t^12+6*t^11+8*t^10+10*t^9+9*t^8+10*t^7+8*t^6+6*t^5+4*t^4+2*t^3

[1,6,7,7,21,42],
d := 42; A:=[1,6,7,7,21]; B := []; B:=[[3,1,1,1]];
D := P(d,A) - PA(A) - Pb(B) - Ppc(7,1,2);
 // t^13+2*t^12+3*t^11+4*t^10+5*t^9+3*t^8+5*t^7+4*t^6+3*t^5+2*t^4+t^3


[4,5,9,9,22,49], b = 1;
E4,E5 = 23,-5, so N = ((23+5)/2+4*1)/r = 2;


========================

Ex. Compose an ice-cream story to prove that
> t^4*Qorb(13,[8],-8);
(t^12 + t^10 + t^7 + t^4 + t^2)/(t^14 - t^13 - t + 1)

is equal to minus

> 1/t^4*Qorb(13,[5],8);
(-t^12 - t^10 - t^7 - t^4 - t^2)/(t^14 - t^13 - t + 1)

========================

The terms in BSz's formula





function First(r,k)
 return &+[ ikbar*(r-ikbar)*(-i-(r-i)*t^r)*t^i
 where ikbar is (i*k mod r) : i in [1..r-1]] / (2*r*(1-t^r)^2);
end function;

function Second(r,k)
 return &+[ ikbar*(r-ikbar)*(r-2*ikbar)*t^i
 where ikbar is (i*k mod r) : i in [1..r-1]] / (6*r*(1-t^r));
end function;

Each of these needs to be modified by rational multiples of
(t+t^2+t^3)/(1-t)^4 and t/(1-t)^2 to get rid of terms in t and t^2;
after which they are Gor symmetric of degree 0 with the right denom
and support.

modified Second becomes if k < r/2

 Second(r,k)-(r-2*k)*((r-k)*k*(t+t^3)-2*k*(r+2*k)*t^2)/6/r/(1-t)^4;
 Second(r,k)-(r-2*k)*((r-k)*k*t-2*(r-k)*(3*r-2*k)*t^2+k*t^3)/6/r/(1-t)^4;

for k in [1..8] do
Second(r,k)-(r-2*k)*((r-k)*k*(t+t^3)-2*k*(r+2*k)*t^2)/6/r/(1-t)^4
eq t^2/Denom([1,1,1,r]) *
 (t^(r-2)*(1+t^kinv)*InvMod(Denom([kinv,kinv,r-kinv]) div (1-t)^3,Phi(r))
 mod Phi(r)) where kinv is InverseMod(k,r);
end for;

for k in [9..16] do
Second(r,k)-(r-2*k)*((r-k)*k*(t+t^3)-2*(r-k)*(3*r-2*k)*t^2)/6/r/(1-t)^4
eq t^2/Denom([1,1,1,r]) *
 (t^(r-2)*(1+t^kinv)*InvMod(Denom([kinv,kinv,r-kinv]) div (1-t)^3,Phi(r))
 mod Phi(r)) where kinv is InverseMod(k,r);
end for;


Second is "composite": namely it is the sum S1 + S2, where
and S1(r,a) = - S2(r,(r-a)).

function S1(r,k)
 return &+[ ikbar*(r-ikbar)^2*t^i
 where ikbar is (i*k mod r) : i in [1..r-1]] / r;
end function;

function S2(r,k)
 return &+[ -ikbar^2*(r-ikbar)*t^i
 where ikbar is (i*k mod r) : i in [1..r-1]] / r;
end function;

These may be something to do with Qorb(1/r(k,k,r-k)).
Note that S2(r,k) == -S1(r,r-k), so we don't need to program the
second.

> (S1(17,4)-S1(17,13))/(1-t^17)-(468*t-1800*t^2+468*t^3)/17/(1-t)^4;
(96*t^17+96*t^16+42*t^15-36*t^14+90*t^13+96*t^12+72*t^11
 -48*t^10+72*t^9+96*t^8+90*t^7-36*t^6+42*t^5+96*t^4+96*t^3)
 / (t^20-3*t^19+3*t^18-t^17-t^3+3*t^2-3*t+1)

This makes him integral Gor symmetric of right deg., but doesn't
say how to express as sum of reciprocals. Probably have to do that
separately with S1 and S2

 S1(17,4)*(1-t^13)*(1-t^4)^2 mod CyclotomicPolynomial(17);
14*t^4 - 20

so

 S1(17,4)/(1-t^17) - (14*t^4-20)*Qorb(17,[4,4,13],-4);
 (-1636/17*t^3 - 696/17*t^2 + 2716/17*t)/(t^4 - 4*t^3 + 6*t^2 - 4*t + 1)

denom is now (1-t)^4

>  -S1(17,13)*(1-t^4)*(1-t^13)^2 mod CyclotomicPolynomial(17);
-14*t^13 + 20

so

> -S1(17,13)/(1-t^17) - (-14/t^4+20)*Qorb(17,[4,13,13],4);
(2716/17*t^3 - 696/17*t^2 - 1636/17*t)/(t^4 - 4*t^3 + 6*t^2 - 4*t + 1)



 S1(17,4)/(1-t^17) -S1(17,13)/(1-t^17)-(24/17)*(45*t^3 - 58*t^2 + 45*t)/(1-t)^4
eq
 -6*(1/t^2+t^2)*t^2*Qorb(17,[4,4,13],-4);


 1/6*Second(17,4)/(1-t^17) - 4/17*(45*t^3 - 58*t^2 + 45*t)/(1-t)^4
 eq -(1+t^4)*Qorb(17,[4,4,13],-4);


An alternative expression giving denom 1/(1-t)^4 is
> 1/6*Second(17,4)/(1-t^17) - (1+t^13)*Qorb(17,[4,13,13],-13);
   (6*t^12+66*t^11-54*t^10-18*t^9+1488/17*t^7-2208/17*t^6
   +1488/17*t^5-18*t^3-54*t^2+66*t+6)/(t^8-4*t^7+6*t^6-4*t^5+t^4)

(t^12 + 11*t^11 - 9*t^10 - 3*t^9 + 248/17*t^7 - 368/17*t^6 + 248/17*t^5 - 3*t^3 -
   9*t^2 + 11*t + 1)/(t^8 - 4*t^7 + 6*t^6 - 4*t^5 + t^4)


The N term should be (Num = sym poly in [3..r]) / Denom([1,1,1,r]). It
should change to - on exchanging k and r-k. e.g.
   1/6*Second(17,4)/(1-t^17) -(78*t-300*t^2+78*t^3)/17/(1-t)^4;
 (16*t^17+16*t^16+7*t^15-6*t^14+15*t^13+16*t^12+12*t^11-8*t^10
   +12*t^9+16*t^8+15*t^7-6*t^6+7*t^5+16*t^4+16*t^3)
 / (t^20 - 3*t^19 + 3*t^18 - t^17 - t^3 + 3*t^2 - 3*t + 1)

For example, the k = 1 case
 1/6*Second(17,1)/(1-t^17) - (40*t-95*t^2+40*t^3)/17/(1-t)^4 ;
 (t^17 + t^16 + t^15 + t^14 + t^13 + t^12 + t^11 + t^10 + t^9 + t^8 + t^7 + t^6 +
 t^5 + t^4 + t^3)/(t^20 - 3*t^19 + 3*t^18 - t^17 - t^3 + 3*t^2 - 3*t + 1)
> 1/6*Second(17,16)/(1-t^17) - (40*t-95*t^2+40*t^3)/17/(1-t)^4 ;

The k = r-1 case
  1/6*Second(17,16)/(1-t^17) - (-40*t+95*t^2-40*t^3)/17/(1-t)^4;
  (-t^17-t^16-t^15-t^14-t^13-t^12-t^11-t^10-t^9-t^8-t^7
   -t^6-t^5-t^4-t^3)/(t^20-3*t^19+3*t^18-t^17-t^3+3*t^2-3*t+1)

The r = 5, k = 2 or 3 cases:

> 1/6*Second(5,2)/(1-t^5) - (t-6*t^2+t^3)/5/(1-t)^4;
 (3*t^5-2*t^4+3*t^3)/(t^8-3*t^7+3*t^6-t^5-t^3+3*t^2-3*t+1)

1/6*Second(5,3)/(1-t^5)+(t-6*t^2+t^3)/5/(1-t)^4;
 (-3*t^5+2*t^4-3*t^3)/(t^8-3*t^7+3*t^6-t^5-t^3+3*t^2-3*t+1)

Acting in the same way with (17,5) and 7 = 5^-1 seems to work less well:
> Second(17,5)/(1-t^17) - 6*(1+t^7)*Qorb(17,[7,7,10],-7) - (1950*t-2400*t^2+1950\
*t^3)/17/(1-t)^4;
(-96*t^6 - 18*t^5 - 18*t - 96)/(t^5 - 4*t^4 + 6*t^3 - 4*t^2 + t)






in this residue, what are the coefficients?

for k in [1..r-1] do
  1/6*Second(17,k)*Denom([kinv,kinv,r-kinv]) mod Phi(r)
  eq 1+t^kinv mod Phi(r) where kinv is InverseMod(k,r);
end for;

Nterm should be Num/denom(1,1,1,r) with Numerator a sym poly in
[3..r] that is (1+t^k)/((1-t^k)^2(1-t^r-k))

Inr

InvMod(Denom([k,2*k,r-k]), Phi(r))


(1+t^k) * InvMod((1-t^k)^2*(1-t^(r-k)),Phi(r))

want XX so that X*(1-t^k)*(1-t^(2*k)*(1-t^(r-k)) eq 1

> k:=1;t^2*InvMod(t*15*(1-t^k)*(1-t^(2*k))*(1-t^(r-k)),Phi(r));


function Qorb(r,LL,k)
 L := [ Integers() | i : i in LL ]; // this allows empty list
 if (k + &+L) mod r ne 0
  then error "Error: Canonical weight not compatible";
 end if;
  n := #LL;
 Pi := &*[ R | 1-t^i : i in LL];
  h := Degree(GCD(1-t^r, Pi)); // degree of GCD(A,B)
  // -- simpler calc?
 l := Floor((k+n+1)/2+h);
 // If l < 0 we need a kludge to avoid programming
  // Laurent polynomials properly
 de := Maximum(0,Ceiling(-l/r));
  m := l + de*r;
  A := (1-t^r) div (1-t);
  B := Pi div (1-t)^n;
  H,al_throwaway,be:=XGCD(A,t^m*B);
 return t^m*be/(H*(1-t)^n*(1-t^r)*t^(de*r));
end function;

// This function is Nterm, the second term
// sum ikbar * (r-ikbar) * (r-2*ikbar)
// in [Buckley and Szendroi] Cor. 3.3

function Second(r,k)
 return &+[ ikbar*(r-ikbar)*(r-2*ikbar)*t^i
 where ikbar is (i*k mod r) : i in [1..r-1]] / (6*r*(1-t^r));
end function;

// After subtracting off initial term to kill coeff of t and t^2,
// it is Num/Denom([1,1,1,r]), where the numerator is symmetric
// integral supported in [3..r]) that is
// (1+t^kinv)(1-t)^3/(1-t^kinv)^2(1-t^(r-kinv) mod (1-t^r)/(1-t)

for k in [1..r-1] do
Second(r,k)-(r-2*k)*(r-k)*k*(t+t^3)/6/r/(1-t)^4
      +((r-2*k)*2*ka*(r+2*ka)*t^2/6/r/(1-t)^4 where ka is Min(k,r-k))
eq t^2/Denom([1,1,1,r]) *
 (t^(r-2)*(1+t^kinv)*InvMod(Denom([kinv,kinv,r-kinv]) div (1-t)^3,Phi(r))
 mod Phi(r)) where kinv is InverseMod(k,r);
end for;

function Nterm(r,b)
return t^2*
 (t^(r-2)*(1+t^b)*InverseMod(Denom([b,b,r-b]) div (1-t)^3,((1-t^r) div (1-t)))
 mod ((1-t^r) div (1-t)))/Denom([1,1,1,r]);
end function;




 for i in [1..20] do
 r := Random(500); k := Random(r);
 if GCD(r,k) eq 1 then b := InverseMod(k,r);
 r,k,Second(r,k)-(r-2*k)*(r-k)*k*(t+t^3)/6/r/(1-t)^4
   +((r-2*k)*2*ka*(r+2*ka)*t^2/6/r/(1-t)^4 where ka is Min(k,r-k))
 eq Nterm(r,b);
 end if; end for;







Plan of action: Use the functions defined at the end of CY, applied to the list
AASK2. PC(A) is a list of the curve contributions, obtained as partial frac
decomposition. Each should be
  b*Qorb - N*Nterm - b*First




function First(r,k)
 return &+[ ikbar*(r-ikbar)*(-i-(r-i)*t^r)*t^i
 where ikbar is (i*k mod r) : i in [1..r-1]] / (2*r*(1-t^r)^2);
end function;


if k < r-k, need to do
First(r,k) - (-k*(r-k)*(t+t^3)+4*k^2*t^2)/(2*r)/(1-t)^4;

or if k > r-k, need do do
First(r,k) - (-k*(r-k)*(t+t^3)+4*(r-k)^2*t^2)/(2*r)/(1-t)^4;

so in either case

First(r,k) - (-k*(r-k)*(t+t^3)+4*ka^2*t^2)/(2*r)/(1-t)^4 where ka is Min(k,r-k);

Next, take out the leading term in 1/(1-t^r)^2 :

k:=8; First(r,k)-(-k*(r-k)*(t+t^3)+4*ka^2*t^2)/(2*r)/(1-t)^4
 -r*Qorb(r,[b,r-b],r)/(1-t^r)
 where ka is Min(k,r-k) where b is InverseMod(k,r);


function BFirst(r,k)
 return &+[ -i*ikbar*(r-ikbar)*t^i
 where ikbar is (i*k mod r) : i in [1..r-1]] / (2*r*(1-t^r));
end function;