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

%I A090351 #25 May 26 2025 09:56:41
%S A090351 1,1,3,15,108,1032,12388,179572,3052986,59555338,1310677726,
%T A090351 32114051862,866766965308,25547102523604,816335926158372,
%U A090351 28107705687291892,1037367351120788551,40852168787823027351,1709792654612819858341
%N A090351 G.f. satisfies A^3 = BINOMIAL(A^2).
%C A090351 In general, if A^n = BINOMIAL(A^(n-1)), then for all integer m>0 there exists an integer sequence B such that B^d = BINOMIAL(A^m) where d=gcd(m+1,n). Also, coefficients of A(k*x)^n = k-th binomial transform of coefficients in A(k*x)^(n-1) for all k>0.
%H A090351 Vaclav Kotesovec, <a href="/A090351/b090351.txt">Table of n, a(n) for n = 0..390</a>
%F A090351 G.f. satisfies: A(x)^3 = A(x/(1-x))^2 / (1-x).
%F A090351 a(n) ~ (n-1)! / (6 * (log(3/2))^(n+1)). - _Vaclav Kotesovec_, Nov 18 2014
%F A090351 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-1) = A050351(n) = 1/2*A004123(n+1) for n >= 1. - _Peter Bala_, May 26 2015
%F A090351 G.f. satisfies [x^n] 1/A(x)^(2*n-2) = [x^n] 1/A(x)^(3*n-3) = -(n-1)*A088222(n) for n >= 0. - _Paul D. Hanna_, Apr 28 2025
%F A090351 G.f.: Product_{k>=1} 1/(1 - k*x)^((1/6) * (2/3)^k). - _Seiichi Manyama_, May 26 2025
%e A090351 A^3 = BINOMIAL(A090352), since A090352=A^2.
%t A090351 nmax = 18; sol = {a[0] -> 1};
%t A090351 Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^3 - A[x/(1 - x)]^2/(1 - x) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t A090351 sol /. Rule -> Set;
%t A090351 a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 02 2019 *)
%t A090351 With[{m=40}, CoefficientList[Series[Exp[Sum[Sum[2^(j-1)*j!* StirlingS2[k,j], {j,k}]*x^k/k, {k,m+1}]], {x,0,m}], x]] (* _G. C. Greubel_, Jun 08 2023 *)
%o A090351 (PARI) {a(n) = my(A); if(n<0,0,A = 1+x +x*O(x^n); for(k=1,n, B = subst(A^2,x,x/(1-x))/(1-x) +x*O(x^n); A = A - A^3 + B); polcoef(A,n,x))}
%o A090351 for(n=0,25,print1(a(n),", "))
%o A090351 (Magma)
%o A090351 m:=40;
%o A090351 f:= func< n,x | Exp((&+[(&+[2^(j-1)*Factorial(j)* StirlingSecond(k,j)*x^k/k: j in [1..k]]): k in [1..n+2]])) >;
%o A090351 R<x>:=PowerSeriesRing(Rationals(), m+1);  // A090351
%o A090351 Coefficients(R!( f(m,x) )); // _G. C. Greubel_, Jun 08 2023
%o A090351 (SageMath)
%o A090351 m=50
%o A090351 def f(n, x): return exp(sum(sum(2^(j-1)*factorial(j)* stirling_number2(k,j)*x^k/k for j in range(1,k+1)) for k in range(1,n+2)))
%o A090351 def A090351_list(prec):
%o A090351     P.<x> = PowerSeriesRing(QQ, prec)
%o A090351     return P( f(m,x) ).list()
%o A090351 A090351_list(m-9) # _G. C. Greubel_, Jun 08 2023
%Y A090351 Cf. A004123, A050351, A084784, A090352, A090353, A090356, A090358, A088222.
%K A090351 nonn,easy
%O A090351 0,3
%A A090351 _Paul D. Hanna_, Nov 26 2003