cp's OEIS Frontend

A370019 Expansion of [ Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(1/3).

%I A370019 #14 Feb 24 2024 05:43:31
%S A370019 1,-4,-16,-48,-384,-2816,-24384,-206336,-1815552,-16189440,-146777856,
%T A370019 -1346648064,-12487131136,-116810932224,-1101080592384,
%U A370019 -10447586845696,-99706199973888,-956400813293568,-9215587975397376,-89158545637244928,-865730439117078528,-8433936444598677504
%N A370019 Expansion of [ Sum_{n>=0} (-4)^n * (2*4^n + 1)/3 * x^(n*(n+1)/2) ]^(1/3).
%C A370019 The self-convolution cube equals A370018.
%H A370019 Paul D. Hanna, <a href="/A370019/b370019.txt">Table of n, a(n) for n = 0..630</a>
%F A370019 a(n) ~ c * d^n / n^(4/3), where d = 10.39336299855957350315151176284030870108168399888... and c = -0.218294054014127126766352511836393819909572679... - _Vaclav Kotesovec_, Feb 24 2024
%e A370019 G.f.: 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 - 146777856*x^10 + ...
%e A370019 RELATED SERIES.
%e A370019 The cube of the g.f. A(x) equals the g.f. A370018 which starts as
%e A370019 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 A370019 The reciprocal of the g.f. A(x) equals the g.f. of A370044, which begins
%e A370019 1/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 + ... + A370044(n)*x^n + ...
%o A370019 (PARI) {a(n) = my(A);
%o A370019 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 A370019 polcoeff(H=A, n)}
%o A370019 for(n=0, 20, print1(a(n), ", "))
%Y A370019 Cf. A370016, A370044, A370336.
%K A370019 sign
%O A370019 0,2
%A A370019 _Paul D. Hanna_, Feb 23 2024