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A090362 G.f. satisfies A^6 = BINOMIAL(A)^5 and also equals A090358^5.

%I A090362 #23 May 26 2025 09:52:40
%S A090362 1,5,40,460,7220,148276,3831760,120333680,4460572870,190679906990,
%T A090362 9230084185456,498734395394840,29740372199558420,1939241402832412180,
%U A090362 137222625361036807760,10470376552560151801616,856818090423771231257245
%N A090362 G.f. satisfies A^6 = BINOMIAL(A)^5 and also equals A090358^5.
%C A090362 See comments in A090358.
%H A090362 Vaclav Kotesovec, <a href="/A090362/b090362.txt">Table of n, a(n) for n = 0..320</a>
%F A090362 G.f.: A(x)^6 = A(x/(1-x))^5/(1-x)^5.
%F A090362 a(n) ~ (n-1)! / (6 * (log(6/5))^(n+1)). - _Vaclav Kotesovec_, Nov 19 2014
%F A090362 From _Peter Bala_, May 26 2015: (Start)
%F A090362 O.g.f.: A(x) = exp( Sum_{n >= 1} b(n)*x^n/n ), where b(n) = Sum_{k = 1..n} k!*Stirling2(n,k)*5^k = A094418(n).
%F A090362 BINOMIAL(A(x)) = exp( Sum_{n >= 1} c(n)*x^n/n ) where c(n) = (-1)^n*Sum_{k = 1..n} k!*Stirling2(n,k)*(-6)^k. A(x) = B(x)^5 and BINOMIAL(A(x)) = B(x)^6 where B(x) = 1 + x + 6*x^2 + 66*x^3 + 1071*x^4 + ... is the o.g.f. for A090358. See also A019538. (End)
%F A090362 G.f.: Product_{k>=1} 1/(1 - k*x)^((1/6) * (5/6)^k). - _Seiichi Manyama_, May 26 2025
%t A090362 nmax = 16; sol = {a[0] -> 1};
%t A090362 Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^6 - A[x/(1 - x)]^5/(1 - x)^5 + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t A090362 sol /. Rule -> Set;
%t A090362 a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 02 2019 *)
%o A090362 (PARI) {a(n)=local(A); if(n<0,0,A=1+x+x*O(x^n); for(k=1,n,B=subst(A,x,x/(1-x))/(1-x)+x*O(x^n); A=A-A^6+B^5);polcoeff(A,n,x))}
%Y A090362 Cf. A090358, A019538, A094418.
%Y A090362 Cf. A090352, A090355, A090357.
%K A090362 nonn,easy
%O A090362 0,2
%A A090362 _Paul D. Hanna_, Nov 26 2003