A370336 Expansion of [ Sum_{n>=0} 5^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(1/3).

1, 5, -25, 300, -3000, 34375, -426750, 5539375, -73968750, 1010175000, -14043011250, 198006675000, -2824523453125, 40684553625000, -590871274218750, 8642318714253125, -127185323309250000, 1881843237600000000, -27976771190059687500, 417688301999460937500, -6259735680122821875000

Offset

0,2

Comments

Self-convolution cube equals A370335.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..630

Formula

a(n) ~ (-1)^(n+1) * c * d^n / n^(4/3), where d = 16.061038491618401040959460250524051290971925631740259277535... and c = 0.25648790376068702946627569573532916303248367815529074... - Vaclav Kotesovec, Feb 24 2024

Example

G.f.: A(x) = 1 + 5*x - 25*x^2 + 300*x^3 - 3000*x^4 + 34375*x^5 - 426750*x^6 + 5539375*x^7 - 73968750*x^8 + 1010175000*x^9 - 14043011250*x^10 + ...
RELATED SERIES.
The cube of the g.f. A(x) yields the g.f. of A370335 starting as
A(x)^3 = 1 + 15*x + 275*x^3 + 5375*x^6 + 106875*x^10 + 2134375*x^15 + 42671875*x^21 + 853359375*x^28 + 17066796875*x^36 + ... + 5^n*(2*4^n + 1)/3 * x^(n*(n+1)/2) + ...

Prog

(PARI) {a(n) = my(A);
A = sum(m=0, sqrtint(2*n+1), 5^m*(2*4^m + 1)/3 * x^(m*(m+1)/2) +x*O(x^n))^(1/3);
polcoeff(H=A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A370335, A370016.

Sequence in context: A145076 A185063 A165656 * A145773 A197195 A358545

Adjacent sequences: A370333 A370334 A370335 * A370337 A370338 A370339

Keyword

sign

Author

Paul D. Hanna, Feb 23 2024

Status

approved