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A264408 E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^3).

%I A264408 #17 Nov 21 2021 05:30:33
%S A264408 1,1,8,123,2880,91380,3670422,178637046,10220188584,672297233904,
%T A264408 49998246211440,4148400045354048,379922492043352728,
%U A264408 38069325641907527760,4143052833221549219112,486645149764644576491160,61364015736691449905270208,8267876533375590035644494336,1185420892076469924434688605760
%N A264408 E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^3).
%H A264408 Seiichi Manyama, <a href="/A264408/b264408.txt">Table of n, a(n) for n = 0..331</a>
%F A264408 E.g.f. also satisfies:
%F A264408 (1) A(x) = ( LambertW(3*log(1-x)) / (3*log(1-x)) )^(1/3).
%F A264408 (2) A(x) = Sum_{n>=0} (3*n+1)^(n-1) * (-log(1-x))^n / n!.
%F A264408 (3) A(x) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (A(x)^3 + k).
%F A264408 (4) A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} |Stirling1(n, k)| * A(x)^(3*k).
%F A264408 a(n) = Sum_{k=0..n} |Stirling1(n, k)| * (3*k+1)^(k-1).
%F A264408 a(n) ~ n^(n-1) / (sqrt(3) * exp(n - n*exp(-1)/3 - 5/6) * (exp(exp(-1)/3) - 1)^(n-1/2)). - _Vaclav Kotesovec_, Nov 18 2015
%e A264408 E.g.f.: A(x) = 1 + x + 8*x^2/2! + 123*x^3/3! + 2880*x^4/4! + 91380*x^5/5! + 3670422*x^6/6! + 178637046*x^7/7! + 10220188584*x^8/8! +...
%e A264408 where A(x) = 1/(1-x)^(A(x)^3).
%e A264408 From a LambertW identity,
%e A264408 A(x) = 1 - log(1-x) + 7*log(1-x)^2/2! - 10^2*log(1-x)^3/3! + 13^3*log(1-x)^4/4! - 16^4*log(1-x)^5/5! + 19^5*log(1-x)^6/6! +...
%e A264408 Also,
%e A264408 A(x) = 1 + x*A(x)^3 + x^2*A(x)^3*(A(x)^3+1)/2! + x^3*A(x)^3*(A(x)^3+1)*(A(x)^3+2)/3! + x^4*A(x)^3*(A(x)^3+1)*(A(x)^3+2)*(A(x)^3+3)/4! +...
%t A264408 Table[Sum[Abs[StirlingS1[n, k]] * (3*k+1)^(k-1), {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Nov 18 2015 *)
%o A264408 (PARI) {a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, x^m/m! * prod(k=0, m-1, A^3 + k) +x*O(x^n)) ); n!*polcoeff(A, n)}
%o A264408 for(n=0, 20, print1(a(n), ", "))
%o A264408 (PARI) a(n) = sum(k=0, n, abs(stirling(n, k, 1))*(3*k+1)^(k-1));
%o A264408 for(n=0, 20, print1(a(n), ", "))
%Y A264408 Cf. A052813, A264407, A349505.
%K A264408 nonn
%O A264408 0,3
%A A264408 _Paul D. Hanna_, Nov 18 2015