A357401 Coefficients in the power series expansion of 1/Sum_{n=-oo..+oo} n * x^(2*n+1) * (1 - x^n)^(n+1).

1, 0, 1, 0, -2, 8, -14, 16, -7, -24, 103, -232, 334, -256, -211, 1400, -3562, 6048, -6470, 512, 17788, -53720, 102983, -134832, 76147, 187960, -776169, 1690880, -2558499, 2270952, 1214672, -10443024, 26674201, -45822896, 51953043, -11147384, -126256811, 401311496

Offset

1,5

Comments

Equals column 1 of triangle A357400. The g.f. G(x,y) of triangle A357400 satisfies: y = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * G(x,y)^n; this sequence gives the coefficients of x^n*y in G(x,y) for n >= 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1500

Formula

G.f. A(x,y) = Sum_{n>=1} a(n) * x^n satisfies the following relations.

(1) A(x) = 1/Sum_{n=-oo..+oo} n * x^(2*n+1) * (1 - x^n)^(n+1).

(2) A(X) = -x/Sum_{n=-oo..+oo} n * (-1)^n * x^(n*(n-1)) / (1 - x^(n+1))^n.

Example

G.f.: A(x) = x + x^3 - 2*x^5 + 8*x^6 - 14*x^7 + 16*x^8 - 7*x^9 - 24*x^10 + 103*x^11 - 232*x^12 + 334*x^13 - 256*x^14 - 211*x^15 + 1400*x^16 - 3562*x^17 + 6048*x^18 - 6470*x^19 + 512*x^20 + ...
Related series.
x/A(x) = 1 - x^2 + 3*x^4 - 8*x^5 + 9*x^6 - 10*x^8 + 24*x^10 - 24*x^11 + 15*x^14 + 9*x^16 - 80*x^17 + 90*x^18 - 43*x^20 + 57*x^22 + ... + A357406(n)*x^n + ...
which equals x*Sum_{n=-oo..+oo} n * x^(2*n+1) * (1 - x^n)^(n+1).

Prog

(PARI) {a(n) = my(A = 1/sum(m=-n-1, n+1, m * x^(2*m+1) * (1 - x^m +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))
(PARI) /* As Column 1 of triangle A357400 (slow) */
{A357400(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=1, 40, print1(A357400(n, 1), ", "))

Crossrefs

Cf. A357400, A357406.

Sequence in context: A349818 A329453 A319964 * A002248 A333968 A194278

Adjacent sequences: A357398 A357399 A357400 * A357402 A357403 A357404

Keyword

sign

Author

Paul D. Hanna, Sep 26 2022

Status

approved