A370045 Expansion of 1 / Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2).

1, 12, 144, 1552, 16512, 172800, 1803200, 18765312, 195167232, 2028914688, 21089678592, 219201730560, 2278287884288, 23679245377536, 246107817345024, 2557891149933568, 26585106479751168, 276308723697205248, 2871777147680423936, 29847423508786839552, 310215112347152351232

Offset: 0

Author

Paul D. Hanna, Feb 24 2024

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 12*x + 144*x^2 + 1552*x^3 + 16512*x^4 + 172800*x^5 + 1803200*x^6 + 18765312*x^7 + 195167232*x^8 + 2028914688*x^9 + 21089678592*x^10 + ...
RELATED SERIES.
The expansion of 1/A(x) is the following series (A370018)
1/A(x) = 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) + ...
The cube root of A(x) begins
A(x)^(1/3) = 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 + ... + A370044(n)*x^n + ...
Also, the sixth root of A(x) is an integer series starting as
A(x)^(1/6) = 1 + 2*x + 14*x^2 + 92*x^3 + 742*x^4 + 6188*x^5 + 54956*x^6 + 498584*x^7 + 4625478*x^8 + 43493324*x^9 + 413627172*x^10 + ...
		

Links

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

Crossrefs

Cf. A370018 (1/A(x)), A370044 (A(x)^(1/3)), A370019 (A(x)^(-1/3)).

Programs

PARI

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

Formula

From Vaclav Kotesovec, Feb 25 2024: (Start)

a(n) ~ c * d^n, where

d = 10.39336299855957350315151176284030870108168399888817592486381041027988779...

c = 1.433973222898078483437999597179822040398973315396494951383570608840342399...

d = 1/r, where r = 0.09621524814812982023560791941974657613430770687333255066... is the smallest positive root of the equation Sum_{k>=0} (-4)^k * (2*4^k + 1) * r^(k*(k+1)/2) = 0. (End)