This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A251586 #14 Sep 08 2022 08:46:10
%S A251586 1,1,8,156,5160,245976,15450912,1209613824,113666333184,
%T A251586 12479546880000,1568823886181376,222308476014034944,
%U A251586 35069155573323036672,6096327654732137496576,1158040133351856000000000,238674982804212474577944576,53050036437721656891731017728,12649916782354997981599305302016
%N A251586 a(n) = 6^(n-4) * (n+1)^(n-6) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296).
%H A251586 G. C. Greubel, <a href="/A251586/b251586.txt">Table of n, a(n) for n = 0..306</a>
%F A251586 Let G(x) = 1 + x*G(x)^6 be the g.f. of A002295, then the e.g.f. A(x) of this sequence satisfies:
%F A251586 (1) A(x) = exp( 6*x*A(x) * G(x*A(x))^5 ) / G(x*A(x))^5.
%F A251586 (2) A(x) = F(x*A(x)) where F(x) = exp(6*x*G(x)^5)/G(x)^5 is the e.g.f. of A251576.
%F A251586 (3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251576.
%F A251586 E.g.f.: -LambertW(-6*x) * (6 + LambertW(-6*x))^5 / (x*6^6). - _Vaclav Kotesovec_, Dec 07 2014
%e A251586 E.g.f.: A(x) = 1 + x + 8*x^2/2! + 156*x^3/3! + 5160*x^4/4! + 245976*x^5/5! +...
%e A251586 such that A(x) = exp( 6*x*A(x) * G(x*A(x))^5 ) / G(x*A(x))^5
%e A251586 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295:
%e A251586 G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
%e A251586 RELATED SERIES.
%e A251586 Note that A(x) = F(x*A(x)) where F(x) = exp(6*x*G(x)^5)/G(x)^5,
%e A251586 F(x) = 1 + x + 6*x^2/2! + 96*x^3/3! + 2736*x^4/4! + 115056*x^5/5! +...
%e A251586 is the e.g.f. of A251576.
%t A251586 Table[6^(n - 4)*(n + 1)^(n - 6)*(125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296), {n, 0, 50}] (* _G. C. Greubel_, Nov 13 2017 *)
%o A251586 (PARI) {a(n) = 6^(n-4) * (n+1)^(n-6) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296)}
%o A251586 for(n=0,20,print1(a(n),", "))
%o A251586 (PARI) {a(n) = local(G=1,A=1); for(i=1,n, G=1+x*G^6 +x*O(x^n));
%o A251586 for(i=1,n, A = exp(6*x*A * subst(G^5,x,x*A) ) / subst(G^5,x,x*A) ); n!*polcoeff(A, n)}
%o A251586 for(n=0, 20, print1(a(n), ", "))
%o A251586 (Magma) [6^(n - 4)*(n + 1)^(n - 6)*(125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296): n in [0..50]]; // _G. C. Greubel_, Nov 13 2017
%Y A251586 Cf. A251576, A002295.
%Y A251586 Cf. Variants: A251583, A251584, A251585, A251587, A251588, A251589, A251590.
%K A251586 nonn
%O A251586 0,3
%A A251586 _Paul D. Hanna_, Dec 06 2014