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%I A251694 #12 Sep 08 2022 08:46:10
%S A251694 1,3,80,4480,389376,46137344,6939332608,1266556600320,272000000000000,
%T A251694 67204714785144832,18780742521990414336,5857307089452073484288,
%U A251694 2016866466756967373209600,759982437118771200000000000,311070869724989874190180941824,137440072511222468264810285891584
%N A251694 a(n) = (2*n+1) * (3*n+1)^(n-2) * 4^n.
%H A251694 G. C. Greubel, <a href="/A251694/b251694.txt">Table of n, a(n) for n = 0..283</a>
%F A251694 Let G(x) = 1 + x*G(x)^4 be the g.f. of A002293, then the e.g.f. A(x) of this sequence satisfies:
%F A251694 (1) A(x) = exp( 4*x*A(x)^3 * G(x*A(x)^3)^3 ) / G(x*A(x)^3).
%F A251694 (2) A(x) = F(x*A(x)^3) where F(x) = exp(4*x*G(x)^3)/G(x) is the e.g.f. of A251664.
%F A251694 (3) A(x) = ( Series_Reversion( x*G(x)^3 / exp(12*x*G(x)^3) )/x )^(1/3).
%F A251694 E.g.f.: (-LambertW(-12*x)/(12*x))^(1/3) * (1 + LambertW(-12*x)/12). - _Vaclav Kotesovec_, Dec 07 2014
%e A251694 E.g.f.: A(x) = 1 + 3*x + 80*x^2/2! + 4480*x^3/3! + 389376*x^4/4! + 46137344*x^5/5! +...
%e A251694 such that A(x) = exp( 4*x*A(x)^3 * G(x*A(x)^3)^3 ) / G(x*A(x)^3),
%e A251694 where G(x) = 1 + x*G(x)^4 is the g.f. A002293:
%e A251694 G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
%e A251694 Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^3) where
%e A251694 F(x) = 1 + 3*x + 26*x^2/2! + 430*x^3/3! + 10872*x^4/4! + 373664*x^5/5! +...
%e A251694 F(x) = exp( 4*x*G(x)^3 ) / G(x) is the e.g.f. of A251664.
%t A251694 Table[(2*n + 1)*(3*n + 1)^(n - 2)*4^n, {n, 0, 50}] (* _G. C. Greubel_, Nov 13 2017 *)
%o A251694 (PARI) {a(n) = (2*n+1) * (3*n+1)^(n-2) * 4^n}
%o A251694 for(n=0,20,print1(a(n),", "))
%o A251694 (PARI) {a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^4 +x*O(x^n));
%o A251694 A = ( serreverse( x*G^3 / exp(12*x*G^3) )/x )^(1/3); n!*polcoeff(A, n)}
%o A251694 for(n=0, 20, print1(a(n), ", "))
%o A251694 (Magma) [ (2*n + 1)*(3*n + 1)^(n - 2)*4^n: n in [0..50]]; // _G. C. Greubel_, Nov 13 2017
%Y A251694 Cf. A251664, A002293.
%Y A251694 Cf. Variants: A127670, A251693, A251695, A251696, A251697, A251698, A251699, A251700.
%K A251694 nonn
%O A251694 0,2
%A A251694 _Paul D. Hanna_, Dec 07 2014

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