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A113669 Self-convolution cube equals A113663, where a(n) = n*A113663(n-1) for n>=1, with a(0)=1.

%I A113669 #15 Jul 25 2021 20:58:52
%S A113669 1,1,6,63,904,16080,337374,8107743,218940480,6554205342,215319184860,
%T A113669 7701064928370,297912862462680,12396725926132990,552257670588677214,
%U A113669 26229243983909050215,1323230977463353055616,70673562984581535191094
%N A113669 Self-convolution cube equals A113663, where a(n) = n*A113663(n-1) for n>=1, with a(0)=1.
%H A113669 Vaclav Kotesovec, <a href="/A113669/b113669.txt">Table of n, a(n) for n = 0..350</a>
%F A113669 G.f. A(x) satisfies:
%F A113669 (1) A(x) = 1 + x*d/dx[x*A(x)^3],
%F A113669 (2) [x^n] exp( x*A(x)^3 ) * (n + 1 - A(x)) = 0 for n > 0,
%F A113669 (3) [x^n] exp( n * x*A(x)^3 ) * (2 - A(x)) = 0 for n > 0. - _Paul D. Hanna_, May 27 2018
%F A113669 From _Vaclav Kotesovec_, Oct 23 2020: (Start)
%F A113669 a(n) ~ c * 3^n * n! * n^(2/3), where c = 0.2509528330393045762351289...
%F A113669 a(n) ~ A113663(n)/3. (End)
%F A113669 a(0) = 1; a(n) = n * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1). - _Ilya Gutkovskiy_, Jul 25 2021
%o A113669 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=1,n, A=1+x*deriv(x*A^3));polcoeff(A,n,x)}
%Y A113669 Cf. A113663, A000699, A113670, A113671, A113672, A113673, A113674.
%K A113669 nonn
%O A113669 0,3
%A A113669 _Paul D. Hanna_, Nov 04 2005