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

%I A370044 #13 Feb 25 2024 05:01:30
%S A370044 1,4,32,240,2048,17920,163904,1526784,14473216,138743808,1342326528,
%T A370044 13078851584,128177979392,1262257356800,12481163427840,
%U A370044 123845494105088,1232601926811648,12300407336042496,123037059803447296,1233275751577944064,12385053557486911488,124585853452251328512
%N A370044 Expansion of [ Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(-1/3).
%H A370044 Paul D. Hanna, <a href="/A370044/b370044.txt">Table of n, a(n) for n = 0..1035</a>
%F A370044 a(n) ~ c * d^n / n^(2/3), where d = 10.3933629985595735031515117628403087010816839988881759248638104... and c = 0.42093748110527419326289922348630166534660617909266766696... - _Vaclav Kotesovec_, Feb 24 2024
%e A370044 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 + ...
%e A370044 RELATED SERIES.
%e A370044 The cube of 1/A(x) equals the g.f. A370018 which starts as
%e A370044 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) + ...
%e A370044 and 1/A(x) equals the g.f. of A370019, which begins
%e A370044 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 + ...
%o A370044 (PARI) {a(n) = my(A);
%o A370044 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);
%o A370044 polcoeff(H=A, n)}
%o A370044 for(n=0, 25, print1(a(n), ", "))
%Y A370044 Cf. A370019, A370018, A370045.
%K A370044 nonn
%O A370044 0,2
%A A370044 _Paul D. Hanna_, Feb 24 2024