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A251585 a(n) = 5^(n-3) * (n+1)^(n-5) * (16*n^3 + 87*n^2 + 172*n + 125).

%I A251585 #12 Sep 08 2022 08:46:10
%S A251585 1,1,7,116,3229,129000,6776875,443200000,34766465625,3185000000000,
%T A251585 333992093359375,39470976000000000,5192072114658203125,
%U A251585 752537122540000000000,119176291179656982421875,20476256583680000000000000,3793880513498167242431640625,754086862404270000000000000000
%N A251585 a(n) = 5^(n-3) * (n+1)^(n-5) * (16*n^3 + 87*n^2 + 172*n + 125).
%H A251585 G. C. Greubel, <a href="/A251585/b251585.txt">Table of n, a(n) for n = 0..312</a>
%F A251585 Let G(x) = 1 + x*G(x)^5 be the g.f. of A002294, then the e.g.f. A(x) of this sequence satisfies:
%F A251585 (1) A(x) = exp( 5*x*A(x) * G(x*A(x))^4 ) / G(x*A(x))^4.
%F A251585 (2) A(x) = F(x*A(x)) where F(x) = exp(5*x*G(x)^4)/G(x)^4 is the e.g.f. of A251575.
%F A251585 (3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251575.
%F A251585 E.g.f.: -LambertW(-5*x) * (5 + LambertW(-5*x))^4 / (x*5^5). - _Vaclav Kotesovec_, Dec 07 2014
%e A251585 E.g.f.: A(x) = 1 + x + 7*x^2/2! + 116*x^3/3! + 3229*x^4/4! + 129000*x^5/5! + ...
%e A251585 such that A(x) = exp( 5*x*A(x) * G(x*A(x))^4 ) / G(x*A(x))^4
%e A251585 where G(x) = 1 + x*G(x)^5 is the g.f. of A002294:
%e A251585 G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 + ...
%e A251585 RELATED SERIES.
%e A251585 Note that A(x) = F(x*A(x)) where F(x) = exp(5*x*G(x)^4)/G(x)^4,
%e A251585 F(x) = 1 + x + 5*x^2/2! + 65*x^3/3! + 1505*x^4/4! + 51505*x^5/5! + ...
%e A251585 is the e.g.f. of A251575.
%t A251585 Table[5^(n - 3)*(n + 1)^(n - 5)*(16*n^3 + 87*n^2 + 172*n + 125), {n, 0, 50}] (* _G. C. Greubel_, Nov 13 2017 *)
%o A251585 (PARI) {a(n) = 5^(n-3) * (n+1)^(n-5) * (16*n^3 + 87*n^2 + 172*n + 125)}
%o A251585 for(n=0,20,print1(a(n),", "))
%o A251585 (PARI) {a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^5 +x*O(x^n));
%o A251585 for(i=1,n, A = exp(5*x*A * subst(G^4,x,x*A) ) / subst(G^4,x,x*A) ); n!*polcoeff(A, n)}
%o A251585 for(n=0, 20, print1(a(n), ", "))
%o A251585 (Magma) [5^(n - 3)*(n + 1)^(n - 5)*(16*n^3 + 87*n^2 + 172*n + 125): n in [0..50]]; // _G. C. Greubel_, Nov 13 2017
%Y A251585 Cf. A251575, A002294.
%Y A251585 Cf. Variants: A251583, A251584, A251586, A251587, A251588, A251589, A251590.
%K A251585 nonn
%O A251585 0,3
%A A251585 _Paul D. Hanna_, Dec 06 2014