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%I #19 Sep 08 2022 08:46:10
%S 1,2,27,756,32805,1940598,145746783,13286025000,1425299311881,
%T 175940774387370,24567422246484579,3828825486242232732,
%U 658868122100830078125,124081133675135015343006,25384277097202185803440935,5605841615843732059988768592,1329181093536536811199747015953
%N a(n) = (n+1) * (2*n+1)^(n-2) * 3^n.
%H G. C. Greubel, <a href="/A251693/b251693.txt">Table of n, a(n) for n = 0..305</a>
%F Let G(x) = 1 + x*G(x)^3 be the g.f. of A001764, then the e.g.f. A(x) of this sequence satisfies:
%F (1) A(x) = exp( 3*x*A(x)^2 * G(x*A(x)^2)^2 ) / G(x*A(x)^2).
%F (2) A(x) = F(x*A(x)^2) where F(x) = exp(3*x*G(x)^2)/G(x) is the e.g.f. of A251663.
%F (3) A(x) = sqrt( Series_Reversion( x*G(x)^2 / exp(6*x*G(x)^2) )/x ).
%F E.g.f.: sqrt(-LambertW(-6*x)/(6*x))*(1+LambertW(-6*x)/6). - _Vaclav Kotesovec_, Dec 07 2014
%e E.g.f.: A(x) = 1 + 2*x + 27*x^2/2! + 756*x^3/3! + 32805*x^4/4! +...
%e such that A(x) = exp( 3*x*A(x)^2 * G(x*A(x)^2)^2 ) / G(x*A(x)^2),
%e where G(x) = 1 + x*G(x)^3 is the g.f. A001764:
%e G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
%e Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^2) where
%e F(x) = 1 + 2*x + 11*x^2/2! + 120*x^3/3! + 2061*x^4/4! + 48918*x^5/5! +...
%e F(x) = exp( 3*x*G(x)^2 ) / G(x) is the e.g.f. of A251663.
%t Table[(n + 1)*(2*n + 1)^(n - 2)*3^n, {n, 0, 50}] (* _G. C. Greubel_, Nov 13 2017 *)
%o (PARI) {a(n) = (n+1) * (2*n+1)^(n-2) * 3^n}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^3 +x*O(x^n));
%o A=( serreverse( x*G^2 / exp(6*x*G^2) )/x )^(1/2); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (Magma) [(n + 1)*(2*n + 1)^(n - 2)*3^n: n in [0..50]]; // _G. C. Greubel_, Nov 13 2017
%Y Cf. A251663, A001764.
%Y Cf. Variants: A127670, A251694, A251695, A251696, A251697, A251698, A251699, A251700.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 07 2014
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