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A251583 a(n) = 3^(n-1) * (n+1)^(n-3) * (n+3).

%I A251583 #10 Nov 13 2017 02:56:11
%S A251583 1,1,5,54,945,23328,750141,29859840,1420541793,78732000000,
%T A251583 4986357828309,355459848339456,28178328756432465,2459548529521606656,
%U A251583 234438580086767578125,24233149581890213117952,2700277512299794365456321,322689729227525728790446080,41170357602396483760424637477,5585797616762880000000000000000
%N A251583 a(n) = 3^(n-1) * (n+1)^(n-3) * (n+3).
%H A251583 G. C. Greubel, <a href="/A251583/b251583.txt">Table of n, a(n) for n = 0..330</a>
%F A251583 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 A251583 (1) A(x) = exp( 3*x*A(x) * G(x*A(x))^2 ) / G(x*A(x))^2.
%F A251583 (2) A(x) = F(x*A(x)) where F(x) = exp(3*x*G(x)^2)/G(x)^2 is the e.g.f. of A251573.
%F A251583 (3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251573.
%F A251583 E.g.f.: -LambertW(-3*x) * (3 + LambertW(-3*x))^2 / (27*x). - _Vaclav Kotesovec_, Dec 07 2014
%e A251583 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 54*x^3/3! + 945*x^4/4! + 23328*x^5/5! +...
%e A251583 such that A(x) = exp( 3*x*A(x) * G(x*A(x))^2 ) / G(x*A(x))^2
%e A251583 where G(x) = 1 + x*G(x)^3 is the g.f. of A001764:
%e A251583 G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
%e A251583 RELATED SERIES.
%e A251583 Note that A(x) = F(x*A(x)) where F(x) = exp(3*x*G(x)^2)/G(x)^2,
%e A251583 F(x) = 1 + x + 3*x^2/2! + 21*x^3/3! + 261*x^4/4! + 4833*x^5/5! +...
%e A251583 is the e.g.f. of A251573.
%t A251583 Table[3^(n - 1)*(n + 1)^(n - 3)*(n + 3), {n, 0, 20}] (* _G. C. Greubel_, Nov 12 2017 *)
%o A251583 (PARI) {a(n) = 3^(n-1) * (n+1)^(n-3) * (n+3)}
%o A251583 for(n=0,20,print1(a(n),", "))
%o A251583 (PARI) {a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^3 +x*O(x^n));
%o A251583 for(i=1,n, A = exp(3*x*A * subst(G^2,x,x*A) ) / subst(G^2,x,x*A) ); n!*polcoeff(A, n)}
%o A251583 for(n=0, 20, print1(a(n), ", "))
%Y A251583 Cf. A251573, A001764.
%Y A251583 Cf. Variants: A251584, A251585, A251586, A251587, A251588, A251589, A251590.
%K A251583 nonn
%O A251583 0,3
%A A251583 _Paul D. Hanna_, Dec 06 2014