OFFSET
0,6
COMMENTS
Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
T(n,n) = binomial(2*n+1, n+1)/(2*n+1) = A000108(n) for n >= 0.
T(n+1,n) = 0 for n>= 0.
T(n+2,n) = binomial(2*n-1, n-1) = A001700(n-1) for n >= 1.
T(n+3,n) = 0 for n>= 0.
T(n+1,1) = A357401(n) for n >= 0.
A356783(n) = Sum_{k=0..n} T(n,k), for n >= 0.
A357402(n) = Sum_{k=0..n} T(n,k) * 2^k, for n >= 0.
A357403(n) = Sum_{k=0..n} T(n,k) * 3^k, for n >= 0.
A357404(n) = Sum_{k=0..n} T(n,k) * 4^k, for n >= 0.
A357405(n) = Sum_{k=0..n} T(n,k) * 5^k, for n >= 0.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..5150
FORMULA
G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^n*y^k satisfies the following relations.
(1) y = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x,y)^n.
(2) y*x*A(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x,y)^n ).
(3) -y*x*A(x,y)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x,y)^n / (1 - x^(n+1)*A(x,y))^n.
(4) -y*A(x,y)^3 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x,y) - x^n)^(n+1) / A(x)^n.
(5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x,y))^(n+1) / A(x,y)^n.
(6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x,y)^n / (A(x,y) - x^(n+1))^n.
EXAMPLE
G.f. A(x,y) = 1 + x*y + x^2*(2*y^2) + x^3*(y + 5*y^3) + x^4*(3*y^2 + 14*y^4) + x^5*(-2*y + 10*y^3 + 42*y^5) + x^6*(8*y - 12*y^2 + 35*y^4 + 132*y^6) + x^7*(-14*y + 36*y^2 - 52*y^3 + 126*y^5 + 429*y^7) + x^8*(16*y - 76*y^2 + 148*y^3 - 210*y^4 + 462*y^6 + 1430*y^8) + x^9*(-7*y + 84*y^2 - 354*y^3 + 590*y^4 - 825*y^5 + 1716*y^7 + 4862*y^9) + x^10*(-24*y - 27*y^2 + 416*y^3 - 1565*y^4 + 2322*y^5 - 3199*y^6 + 6435*y^8 + 16796*y^10) + ...
such that
y = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x,y)^2 + x^(-1)/A(x,y) + x*0 + x^3*(1 - x)^2*A(x,y) + x^5*(1 - x^2)^3*A(x,y)^2 + x^7*(1 - x^3)^4*A(x,y)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x,y)^n + ...
also
-y*A(x,y)^3 = ... + x^(-3)*(A(x,y) - x^(-2))^(-1)*A(x,y)^2 + x^(-1)*A(x,y) + x*(A(x,y) - 1) + x^3*(A(x,y) - x)^2/A(x,y) + x^5*(1 - x^2)^3/A(x,y)^2 + x^7*(A(x,y) - x^3)^4/A(x,y)^3 + ... + x^(2*n+1)*(A(x,y) - x^n)^(n+1)/A(x,y)^n + ...
This triangle of coefficients T(n,k) of x^n*y^k, k = 0..n, in g.f. A(x,y) begins:
n = 0: [1],
n = 1: [0, 1],
n = 2: [0, 0, 2],
n = 3: [0, 1, 0, 5],
n = 4: [0, 0, 3, 0, 14],
n = 5: [0, -2, 0, 10, 0, 42],
n = 6: [0, 8, -12, 0, 35, 0, 132],
n = 7: [0, -14, 36, -52, 0, 126, 0, 429],
n = 8: [0, 16, -76, 148, -210, 0, 462, 0, 1430],
n = 9: [0, -7, 84, -354, 590, -825, 0, 1716, 0, 4862],
n = 10: [0, -24, -27, 416, -1565, 2322, -3199, 0, 6435, 0, 16796],
n = 11: [0, 103, -276, -120, 1950, -6732, 9086, -12320, 0, 24310, 0, 58786],
n = 12: [0, -232, 987, -1752, -560, 8832, -28490, 35464, -47268, 0, 92378, 0, 208012],
n = 13: [0, 334, -2160, 6436, -9460, -2673, 39102, -119296, 138294, -180960, 0, 352716, 0, 742900],
n = 14: [0, -256, 3002, -14484, 36218, -46902, -12929, 170368, -495846, 539240, -691900, 0, 1352078, 0, 2674440], ...
in which the main diagonal equals the Catalan numbers (A000108).
PROG
(PARI) {T(n, k) = my(A=[1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff(y - sum(m=-#A\2-1, #A\2+1, x^(2*m+1) * (1 - x^m +x*O(x^#A))^(m+1) * Ser(A)^m ), #A-2); ); polcoeff(A[n+1], k, y)}
for(n=0, 15, for(k=0, n, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Sep 26 2022
STATUS
approved