A370044 Expansion of [ Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(-1/3).

1, 4, 32, 240, 2048, 17920, 163904, 1526784, 14473216, 138743808, 1342326528, 13078851584, 128177979392, 1262257356800, 12481163427840, 123845494105088, 1232601926811648, 12300407336042496, 123037059803447296, 1233275751577944064, 12385053557486911488, 124585853452251328512

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1035

Formula

a(n) ~ c * d^n / n^(2/3), where d = 10.3933629985595735031515117628403087010816839988881759248638104... and c = 0.42093748110527419326289922348630166534660617909266766696... - Vaclav Kotesovec, Feb 24 2024

Example

G.f.: A(x) = 1 + 4*x + 32*x^2 + 240*x^3 + 2048*x^4 + 17920*x^5 + 163904*x^6 + 1526784*x^7 + 14473216*x^8 + 138743808*x^9 + 1342326528*x^10 + ...
RELATED SERIES.
The cube of 1/A(x) equals the g.f. A370018 which starts as
1/A(x)^3 = 1 - 12*x + 176*x^3 - 2752*x^6 + 43776*x^10 - 699392*x^15 + 11186176*x^21 + ... + (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) + ...
and 1/A(x) equals the g.f. of A370019, which begins
1/A(x) = 1 - 4*x - 16*x^2 - 48*x^3 - 384*x^4 - 2816*x^5 - 24384*x^6 - 206336*x^7 - 1815552*x^8 - 16189440*x^9 + ... + A370019(n)*x^n + ...

Prog

(PARI) {a(n) = my(A);
A = sum(m=0, sqrtint(2*n+1), (-4)^m * (1 + 2*4^m)/3 * x^(m*(m+1)/2) +x*O(x^n))^(-1/3);
polcoeff(H=A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A370019, A370018, A370045.

Sequence in context: A301438 A331334 A301779 * A147551 A316803 A299649

Adjacent sequences: A370041 A370042 A370043 * A370045 A370046 A370047

Keyword

nonn

Author

Paul D. Hanna, Feb 24 2024

Status

approved