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A264407 E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^2).

%I A264407 #24 Nov 21 2021 05:30:42
%S A264407 1,1,6,66,1084,23920,665388,22374884,883177328,40043323728,
%T A264407 2051202965280,117166763184768,7384596609153696,509084508866799840,
%U A264407 38108295339435463296,3078340850588419228800,266906341797637061659392,24724454378396015985551616,2436960508983873399401081856,254658073346711773211982974976,28122779871625104764662272952320
%N A264407 E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^2).
%H A264407 Seiichi Manyama, <a href="/A264407/b264407.txt">Table of n, a(n) for n = 0..347</a>
%F A264407 E.g.f. also satisfies:
%F A264407 (1) A(x) = sqrt( LambertW(2*log(1-x)) / (2*log(1-x)) ).
%F A264407 (2) A(x) = Sum_{n>=0} (2*n+1)^(n-1) * (-log(1-x))^n / n!.
%F A264407 (3) A(x) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (A(x)^2 + k).
%F A264407 (4) A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} |Stirling1(n, k)| * A(x)^(2*k).
%F A264407 a(n) = Sum_{k=0..n} |Stirling1(n, k)| * (2*k+1)^(k-1).
%F A264407 a(n) ~ exp(1 + (exp(-1)/2 - 1)*n) * n^(n-1) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n-1/2)). - _Vaclav Kotesovec_, Nov 18 2015
%e A264407 E.g.f.: A(x) = 1 + x + 6*x^2/2! + 66*x^3/3! + 1084*x^4/4! + 23920*x^5/5! + 665388*x^6/6! + 22374884*x^7/7! + 883177328*x^8/8! +...
%e A264407 where A(x) = 1/(1-x)^(A(x)^2).
%e A264407 From a LambertW identity,
%e A264407 A(x) = 1 - log(1-x) + 5*log(1-x)^2/2! - 7^2*log(1-x)^3/3! + 9^3*log(1-x)^4/4! - 11^4*log(1-x)^5/5! + 13^5*log(1-x)^6/6! +...
%e A264407 Also,
%e A264407 A(x) = 1 + x*A(x)^2 + x^2*A(x)^2*(A(x)^2+1)/2! + x^3*A(x)^2*(A(x)^2+1)*(A(x)^2+2)/3! + x^4*A(x)^2*(A(x)^2+1)*(A(x)^2+2)*(A(x)^2+3)/4! +...
%t A264407 Table[Sum[Abs[StirlingS1[n, k]] * (2*k+1)^(k-1), {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Nov 18 2015 *)
%o A264407 (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^2 + k) +x*O(x^n)) ); n!*polcoeff(A, n)}
%o A264407 for(n=0, 20, print1(a(n), ", "))
%o A264407 (PARI) a(n) = sum(k=0,n,abs(stirling(n, k, 1))*(2*k+1)^(k-1));
%o A264407 for(n=0,20,print1(a(n),", "))
%Y A264407 Cf. A052813, A264408, A349504.
%K A264407 nonn
%O A264407 0,3
%A A264407 _Paul D. Hanna_, Nov 17 2015