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A251584 a(n) = 4^(n-2) * (n+1)^(n-4) * (3*n^2 + 13*n + 16).

%I A251584 #15 Sep 08 2022 08:46:10
%S A251584 1,1,6,82,1856,59904,2533888,133169152,8384643072,616038400000,
%T A251584 51781055676416,4903194794655744,516634816527990784,
%U A251584 59967828129860288512,7604226293760000000000,1046004175955626414833664,155145294199098982239567872,24683771056652857103433596928,4193546668531304927540807729152,757730637305085952000000000000000
%N A251584 a(n) = 4^(n-2) * (n+1)^(n-4) * (3*n^2 + 13*n + 16).
%H A251584 G. C. Greubel, <a href="/A251584/b251584.txt">Table of n, a(n) for n = 0..322</a>
%F A251584 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 A251584 (1) A(x) = exp( 4*x*A(x) * G(x*A(x))^3 ) / G(x*A(x))^3.
%F A251584 (2) A(x) = F(x*A(x)) where F(x) = exp(4*x*G(x)^3)/G(x)^3 is the e.g.f. of A251574.
%F A251584 (3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251574.
%F A251584 E.g.f.: -LambertW(-4*x) * (4 + LambertW(-4*x))^3 / (256*x). - _Vaclav Kotesovec_, Dec 07 2014
%e A251584 E.g.f.: A(x) = 1 + x + 6*x^2/2! + 82*x^3/3! + 1856*x^4/4! + 59904*x^5/5! + ...
%e A251584 such that A(x) = exp( 4*x*A(x) * G(x*A(x))^3 ) / G(x*A(x))^3
%e A251584 where G(x) = 1 + x*G(x)^3 is the g.f. of A002293:
%e A251584 G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 + ...
%e A251584 RELATED SERIES.
%e A251584 Note that A(x) = F(x*A(x)) where F(x) = exp(4*x*G(x)^3)/G(x)^3,
%e A251584 F(x) = 1 + x + 4*x^2/2! + 40*x^3/3! + 712*x^4/4! + 18784*x^5/5! + ...
%e A251584 is the e.g.f. of A251574.
%t A251584 Table[4^(n - 2)*(n + 1)^(n - 4)*(3*n^2 + 13*n + 16), {n, 0, 50}] (* _G. C. Greubel_, Nov 13 2017 *)
%o A251584 (PARI) {a(n) = 4^(n-2) * (n+1)^(n-4) * (3*n^2 + 13*n + 16) }
%o A251584 for(n=0,20,print1(a(n),", "))
%o A251584 (PARI) {a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^4 +x*O(x^n));for(i=1,n, A = exp(4*x*A * subst(G^3,x,x*A) ) / subst(G^3,x,x*A) ); n!*polcoeff(A, n)}
%o A251584 for(n=0, 20, print1(a(n), ", "))
%o A251584 (Magma) [4^(n - 2)*(n + 1)^(n - 4)*(3*n^2 + 13*n + 16): n in [0..50]]; // _G. C. Greubel_, Nov 13 2017
%Y A251584 Cf. A251574, A001764.
%Y A251584 Cf. Variants: A251583, A251585, A251586, A251587, A251588, A251589, A251590.
%K A251584 nonn
%O A251584 0,3
%A A251584 _Paul D. Hanna_, Dec 06 2014