a:=2; b:=3; c:=5; // a,b,c any 3 coprime integers
P := [a*b,a*c,b*c]; k := - &+P; k; // work with PP(a*b,a*c,b*c)
X := 1/Denom(P)- Qorb(P[1],[P[2],P[3]],k)- Qorb(P[2],[P[1],P[3]],k)
 - Qorb(P[3],[P[1],P[2]],k); X;
PartialFractionDecomposition(X*(1-t)^3*t^10); // why know it's 10?
Y:=X+1/t^13/Denom([1,2,2]);
PartialFractionDecomposition(Y*(1-t)^3*t^13);
Y*t^13*Denom([1,5,5]);
X eq - 1/t^13/Denom([1,2,2])
  + (1+2*t^2+t^3+2*t^4+t^6)/t^13/Denom([1,5,5]);


we learn from this that PP(6,10,15) has Hilbert series made up of the
sum of

Qorb(1/6(10,15),-31) = 1/s^10(-1 - s^2 - s^3 - s^4 - s^5 - 3*s^6 - s^7
 - 3*s^8 - 3*s^9 - 3*s^10 - 3*s^11 + ..)

Qorb(1/10(6,15),-31) = 1/s^8(-1 + s - 2*s^2 + s^3 - 2*s^4 - s^6 - s^7
 - s^8 - s^9 + ..)

Qorb(1/15(6,10),-31) = 1/s^7(1 + s^3 + s^5 + 2*s^6 + s^8 + ..)

plus

- 1/t^13/&*[1-t^i : i in [1,2,2]] =
1/s^13(-1 - s - 3*s^2 - 3*s^3 - 6*s^4 - 6*s^5 - 10*s^6 - 10*s^7 - 15*s^8
- 15*s^9 - 21*s^10 - 21*s^11 - 28*s^12 - 28*s^13 - 36*s^14 ..)

+ (1+2*t^2+t^3+2*t^4+t^6)/t^13/&*[1-t^i : i in [1,5,5]] =
1/s^13*(1 + s + 3*s^2 + 4*s^3 + 6*s^4 + 8*s^5 + 9*s^6 + 13*s^7 + 15*s^8
+ 19*s^9 + 22*s^10 + 24*s^11 + 30*s^12 + 33*s^13 + 39*s^14


sim, PP(6,14,21)
X eq
-1/t^18/Denom([1,2,2])
+2/t^18/Denom([1,1,3])
+(-t^10-2*t^9-3*t^7-2*t^6-3*t^5-2*t^4-3*t^3-2*t-1)/t^18/Denom([1,7,7]);

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

a:=5; b:=11; c:=13;
P := [a*b,a*c,b*c]; k := - &+P; k; // work with PP(a*b,a*c,b*c)
X := 1/Denom(P)- Qorb(P[1],[P[2],P[3]],k)- Qorb(P[2],[P[1],P[3]],k)
 - Qorb(P[3],[P[1],P[2]],k); X;
W:=PartialFractionDecomposition(X*(1-t)^3*t^115);
A:=W[1,3]/W[1,1];
IsgorensteinSymmetric(X);
IsgorensteinSymmetric(X*t^115*(1-t)^3);
X*t^115*(1-t)^3 - A/t^15;
IsgorensteinSymmetric($1);
X*t^130*(1-t)^3 - A;
W1 := PartialFractionDecomposition($1);
W1;
W1[2,3]/W[2,1]^2;
(W1[2,3]+W1[2,1])/W[2,1]^2;
(W1[2,3]+(1-t)*W1[2,1])/W[2,1]^2;
(W1[2,3]+(1-t+t^3)*W1[2,1])/W[2,1]^2;
(W1[2,3]+(1-t+t^3-t^4)*W1[2,1])/W[2,1]^2;
IsgorensteinSymmetric($1);
B:=(W1[2,3]+(1-t+t^3-t^4)*W1[2,1])/W[2,1]^2;
X*t^130*(1-t)^3 - A - B;
IsgorensteinSymmetric($1);
X*t^130*(1-t)^3 - A - B;
PartialFractionDecomposition($1);
X*t^130*(1-t)^3 - A - B;
W2:=PartialFractionDecomposition($1);
W2[3,3]/W2[3,1]^2;
IsgorensteinSymmetric($1);
(W2[3,3]-3*W2[3,1])/W2[3,1]^2;
(W2[3,3]+(-3+2*t)*W2[3,1])/W2[3,1]^2;
(W2[3,3]+(-3+2*t-2*t^3)*W2[3,1])/W2[3,1]^2;
(W2[3,3]+(-3+2*t-2*t^3+2*t^4)*W2[3,1])/W2[3,1]^2;
(W2[3,3]+(-3+2*t-2*t^3+2*t^4-t^5)*W2[3,1])/W2[3,1]^2;
IsgorensteinSymmetric($1);
C:=(W2[3,3]+(-3+2*t-2*t^3+2*t^4-t^5)*W2[3,1])/W2[3,1]^2;
X*t^130*(1-t)^3 - A - B - C;
W3:=PartialFractionDecomposition($1);
W[3,3]/W[3,1];
IsgorensteinSymmetric($1);
(W[3,3]-W[3,1])/W[3,1];
IsgorensteinSymmetric($1);
B2 := -1/t*(W[3,3]-W[3,1])/W[3,1];
X*t^130*(1-t)^3 - A - B - B2 - C;
PartialFractionDecomposition($1);
X*t^130*(1-t)^3 - A - B - C;
W3:=PartialFractionDecomposition($1);
W3[1,3]/W3[1,1];
IsgorensteinSymmetric($1);
B2:=W3[1,3]/W3[1,1];
X*t^130*(1-t)^3 - A - B - B2 - C;
PartialFractionDecomposition($1);
C2 := X*t^130*(1-t)^3 - A - B - B2 - C;
X*t^130*(1-t)^3 - A - B - B2 - C - C2;
X*t^130 eq A/(1-t)^3 + (B+B2)/(1-t)^3 + (C+C2)/(1-t)^3;
AA := -(3*t^3-t^2+3*t)/t^130;
BB := (B+B2)/(1-t)^3*Denom([1,11,11])/t^130;
CC := (C+C2)/(1-t)^3*Denom([1,13,13])/t^130;
X eq AA/Denom([1,1,5]) + BB/Denom([1,11,11])
 + CC/Denom([1,13,13]);



To draw the conclusion

The Hilbert series of PP(55,65,143) with k = -263
  1/Denom(P)
 // is made of
contributions := 
[
   Qorb(P[1],[P[2],P[3]],k),
   Qorb(P[2],[P[1],P[3]],k),
   Qorb(P[3],[P[1],P[2]],k), // from embedded points
   AA/&*[1-t^i : i in [1,1,5]],
   BB/Denom([1,11,11]),
   CC/Denom([1,13,13])
];

where AA is (-3*t^2 + t - 3)/t^129
BB is (t^18 + t^17 + 2*t^16 + t^15 + 3*t^14 + t^13 + 3*t^12 + 3*t^11 + 3*t^10
 + 4*t^9 + 3*t^8 + 3*t^7 + 3*t^6 + t^5 + 3*t^4 + t^3 + 2*t^2 + t + 1)/t^129
CC is (2*t^22 + t^21 + 3*t^20 + 4*t^19 + 2*t^18 + 7*t^17 + 4*t^16
 + 7*t^15 + 7*t^14 + 6*t^13 + 10*t^12 + 7*t^11 + 10*t^10 + 6*t^9 + 7*t^8
 + 7*t^7 + 4*t^6 + 7*t^5 + 2*t^4 + 4*t^3 + 3*t^2 + t + 2)/t^129

[S! (m*t^129) : m in contributions];
[
    s^14 - s^15 + s^17 - s^18 + 2*s^19 - s^20 + s^22 - s^23 + 2*s^24 - 2*s^26 + 
        2*s^27 + s^29 + s^30 - 2*s^31 + 2*s^32 + s^34 + s^35 - s^36 + s^38 + 
        3*s^39 - s^40 + s^43 + 3*s^44 - s^45 + 2*s^47 - 2*s^48 + 5*s^49 + 
        O(s^50),
    s^17 - s^19 + 2*s^22 - s^24 + 2*s^27 - 2*s^29 + 2*s^30 + s^32 - 2*s^34 + 
        4*s^35 + s^37 - 3*s^39 + 4*s^40 + 3*s^43 - 3*s^44 + 3*s^45 + 6*s^48 - 
        4*s^49 + O(s^50),
    -s^22 - s^23 + s^32 - s^33 - s^34 - 2*s^35 - 2*s^36 + s^42 + s^43 - s^44 - 
        2*s^46 - 2*s^47 - 3*s^48 - 3*s^49 + O(s^50),
    -3 - 5*s - 10*s^2 - 15*s^3 - 20*s^4 - 28*s^5 - 35*s^6 - 45*s^7 - 55*s^8 - 
        65*s^9 - 78*s^10 - 90*s^11 - 105*s^12 - 120*s^13 - 135*s^14 - 153*s^15 -
        170*s^16 - 190*s^17 - 210*s^18 - 230*s^19 - 253*s^20 - 275*s^21 - 
        300*s^22 - 325*s^23 - 350*s^24 - 378*s^25 - 405*s^26 - 435*s^27 - 
        465*s^28 - 495*s^29 - 528*s^30 - 560*s^31 - 595*s^32 - 630*s^33 - 
        665*s^34 - 703*s^35 - 740*s^36 - 780*s^37 - 820*s^38 - 860*s^39 - 
        903*s^40 - 945*s^41 - 990*s^42 - 1035*s^43 - 1080*s^44 - 1128*s^45 - 
        1175*s^46 - 1225*s^47 - 1275*s^48 - 1325*s^49 + O(s^50),
    1 + 2*s + 4*s^2 + 5*s^3 + 8*s^4 + 9*s^5 + 12*s^6 + 15*s^7 + 18*s^8 + 22*s^9 
        + 25*s^10 + 30*s^11 + 35*s^12 + 40*s^13 + 45*s^14 + 52*s^15 + 56*s^16 + 
        63*s^17 + 70*s^18 + 76*s^19 + 84*s^20 + 90*s^21 + 99*s^22 + 108*s^23 + 
        116*s^24 + 125*s^25 + 136*s^26 + 143*s^27 + 154*s^28 + 165*s^29 + 
        174*s^30 + 186*s^31 + 195*s^32 + 208*s^33 + 221*s^34 + 232*s^35 + 
        245*s^36 + 260*s^37 + 270*s^38 + 285*s^39 + 300*s^40 + 312*s^41 + 
        328*s^42 + 340*s^43 + 357*s^44 + 374*s^45 + 388*s^46 + 405*s^47 + 
        424*s^48 + 437*s^49 + O(s^50),
    2 + 3*s + 6*s^2 + 10*s^3 + 12*s^4 + 19*s^5 + 23*s^6 + 30*s^7 + 37*s^8 + 
        43*s^9 + 53*s^10 + 60*s^11 + 70*s^12 + 80*s^13 + 89*s^14 + 102*s^15 + 
        114*s^16 + 125*s^17 + 141*s^18 + 153*s^19 + 170*s^20 + 185*s^21 + 
        199*s^22 + 219*s^23 + 233*s^24 + 253*s^25 + 271*s^26 + 288*s^27 + 
        311*s^28 + 331*s^29 + 351*s^30 + 376*s^31 + 396*s^32 + 423*s^33 + 
        446*s^34 + 468*s^35 + 498*s^36 + 519*s^37 + 549*s^38 + 575*s^39 + 
        600*s^40 + 633*s^41 + 661*s^42 + 690*s^43 + 724*s^44 + 752*s^45 + 
        789*s^46 + 820*s^47 + 850*s^48 + 890*s^49 + O(s^50)
]
> &+$1;
O(s^50)

What is missing is a prediction of what the contributions are as some
kind of ice cream functions.

two dimensional initial term PI = 0
terms from the 1-dim locus all start at t^(-129) degree -129.
This is close to the magic number
  Floor((k+n+1)/2) = 1/2 (-263 + 2 + 1) = -130
or possibly
  Floor((k+n+1)/2) + d = 1/2 (-263 + 2 + 1) + 1 = -129

AA is (-3*t^2 + t - 3)/t^129/[1,1,5] Gor of deg 2-2*129-7 = -263 = k
BB is (t^18 + .. + 1)/t^129/[1,11,11] Gor of deg 18 - 2*129 - 23 = k
CC is (2t^22 + .. + 2)/t^129/[1,13,13] Gor of deg 22 - 2*129 - 27 = k

Now, BB = t^-129 + .. + t^-111 is in range [ga + 1 .. ga + 2r -3] if we take
 ga = -130, r = 11
this works also for CC = 2t^-129 + .. 2t^-107.
If I write AA = (-3 .. -3t^6)/t^129/[1,5,5] it goes in the range -129 .. -123,
which is same statement

The terms from isolated points come from this + 14, 17, 22
55, 65, 143

> X:= (1+2*t^2+t^3+2*t^4+t^6)/t^13/&*[1-t^i : i in [1,5,5]];
> B:=(t^2+1)*Qorb(5,[1],-26)/(1-t^5)/t;
> (X-B)*&*[1-t^i : i in [1,1,5]];
(t^2 - t + 1)/t^13

In other words, the term in X in 1/(1-t^5)^2 is an appropriate
multiple of Qorb(5,[1],-26)/(1-t^5).