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

%I A370045 #11 Feb 25 2024 05:14:57
%S A370045 1,12,144,1552,16512,172800,1803200,18765312,195167232,2028914688,
%T A370045 21089678592,219201730560,2278287884288,23679245377536,
%U A370045 246107817345024,2557891149933568,26585106479751168,276308723697205248,2871777147680423936,29847423508786839552,310215112347152351232
%N A370045 Expansion of 1 / Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2).
%H A370045 Paul D. Hanna, <a href="/A370045/b370045.txt">Table of n, a(n) for n = 0..630</a>
%F A370045 From _Vaclav Kotesovec_, Feb 25 2024: (Start)
%F A370045 a(n) ~ c * d^n, where
%F A370045 d = 10.39336299855957350315151176284030870108168399888817592486381041027988779...
%F A370045 c = 1.433973222898078483437999597179822040398973315396494951383570608840342399...
%F A370045 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)
%e A370045 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 + ...
%e A370045 RELATED SERIES.
%e A370045 The expansion of 1/A(x) is the following series (A370018)
%e A370045 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) + ...
%e A370045 The cube root of A(x) begins
%e A370045 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 + ...
%e A370045 Also, the sixth root of A(x) is an integer series starting as
%e A370045 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 + ...
%o A370045 (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)}
%o A370045 for(n=0, 25, print1(a(n), ", "))
%Y A370045 Cf. A370018 (1/A(x)), A370044 (A(x)^(1/3)), A370019 (A(x)^(-1/3)).
%K A370045 nonn
%O A370045 0,2
%A A370045 _Paul D. Hanna_, Feb 24 2024