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A090352 G.f. satisfies A^3 = BINOMIAL(A)^2, where A = A090351^2.

%I A090352 #27 May 28 2025 07:21:36
%S A090352 1,2,7,36,255,2370,27713,393352,6582068,126888632,2767912036,
%T A090352 67362737168,1808596304964,53083358012760,1690443996202428,
%U A090352 58039582729688320,2136931230333535178,83981145793974066484
%N A090352 G.f. satisfies A^3 = BINOMIAL(A)^2, where A = A090351^2.
%C A090352 See comments in A090351.
%H A090352 G. C. Greubel, <a href="/A090352/b090352.txt">Table of n, a(n) for n = 0..390</a>
%F A090352 G.f. satisfies: A(x)^3 = A(x/(1-x))^2/(1-x)^2.
%F A090352 From _Peter Bala_, May 26 2015: (Start)
%F A090352 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)*2^k = A004123(n+1) = 2*A050351(n) for n >= 1. Cf. A084785.
%F A090352 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)*(-3)^k = A201339(n) = 3*A050351(n) for n >= 1.
%F A090352 A(x) = B(x)^2 and BINOMIAL(A(x)) = B(x)^3 where B(x) = 1 + x + 3*x^2 + 15*x^3 + 108*x^4 + ... is the o.g.f. for A090351. See also A019538. (End)
%F A090352 G.f.: Product_{k>=1} 1/(1 - k*x)^((1/3) * (2/3)^k). - _Seiichi Manyama_, May 26 2025
%F A090352 a(n) ~ (n-1)! / (3 * log(3/2)^(n+1)). - _Vaclav Kotesovec_, May 28 2025
%t A090352 nmax = 17; sol = {a[0] -> 1};
%t A090352 Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^3 - A[x/(1 - x)]^2/(1 - x)^2 + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t A090352 sol /. Rule -> Set;
%t A090352 a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 02 2019 *)
%o A090352 (PARI) {a(n)=local(A); if(n<1,0,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^3+B^2); polcoeff(A,n,x))}
%o A090352 (Magma)
%o A090352 m:=40;
%o A090352 f:= func< n, x | Exp((&+[(&+[2^j*Factorial(j)*StirlingSecond(k, j)*x^k/k: j in [1..k]]): k in [1..n+2]])) >;
%o A090352 R<x>:=PowerSeriesRing(Rationals(), m+1);  // A090352
%o A090352 Coefficients(R!( f(m, x) )); // _G. C. Greubel_, Jul 07 2023
%o A090352 (SageMath)
%o A090352 m=50
%o A090352 def f(n, x): return exp(sum(sum(2^j*factorial(j)*stirling_number2(k, j)*x^k/k for j in range(1, k+1)) for k in range(1, n+2)))
%o A090352 def A090352_list(prec):
%o A090352     P.<x> = PowerSeriesRing(QQ, prec)
%o A090352     return P( f(m, x) ).list()
%o A090352 A090352_list(m-9) # _G. C. Greubel_, Jul 07 2023
%Y A090352 Cf. A004123, A019538, A050351, A084785, A090351, A201339.
%Y A090352 Cf. A090355, A090357, A090362.
%K A090352 nonn,easy
%O A090352 0,2
%A A090352 _Paul D. Hanna_, Nov 26 2003