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a(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n.
9

%I #11 Sep 08 2022 08:46:10

%S 1,5,324,44928,9716112,2870090496,1077194894400,490873123897344,

%T 263285585800098048,162505400851637010432,113463916253636561519616,

%U 88423664876285081860177920,76086820231309990402228260864,71651521268311905104861664903168,73298071049899905319337719679434752

%N a(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n.

%H G. C. Greubel, <a href="/A251696/b251696.txt">Table of n, a(n) for n = 0..256</a>

%F 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 (1) A(x) = exp( 6*x*A(x)^5 * G(x*A(x)^5)^5 ) / G(x*A(x)^5).

%F (2) A(x) = F(x*A(x)^5) where F(x) = exp(6*x*G(x)^5)/G(x) is the e.g.f. of A251666.

%F (3) A(x) = ( Series_Reversion( x*G(x)^5 / exp(30*x*G(x)^5) )/x )^(1/5).

%F E.g.f.: (-LambertW(-30*x)/(30*x))^(1/5) * (1 + LambertW(-30*x)/30). - _Vaclav Kotesovec_, Dec 07 2014

%e E.g.f.: A(x) = 1 + 5*x + 324*x^2/2! + 44928*x^3/3! + 9716112*x^4/4! + 2870090496*x^5/5! +...

%e such that A(x) = exp( 6*x*A(x)^5 * G(x*A(x)^5)^5 ) / G(x*A(x)^5),

%e where G(x) = 1 + x*G(x)^6 is the g.f. A002295:

%e G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...

%e Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^5) where

%e F(x) = 1 + 5*x + 74*x^2/2! + 2028*x^3/3! + 83352*x^4/4! + 4607496*x^5/5! +...

%e F(x) = exp( 6*x*G(x)^5 ) / G(x) is the e.g.f. of A251666.

%t Table[(4*n + 1)*(5*n + 1)^(n - 2)*6^n, {n, 0, 50}] (* _G. C. Greubel_, Nov 14 2017 *)

%o (PARI) {a(n) = (4*n+1) * (5*n+1)^(n-2) * 6^n}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^6 +x*O(x^n));

%o A = ( serreverse( x*G^5 / exp(30*x*G^5) )/x )^(1/5); n!*polcoeff(A, n)}

%o for(n=0, 20, print1(a(n), ", "))

%o (Magma) [(4*n + 1)*(5*n + 1)^(n - 2)*6^n: n in [0..50]]; // _G. C. Greubel_, Nov 14 2017

%Y Cf. A251666, A002295.

%Y Cf. Variants: A127670, A251693, A251694, A251695, A251697, A251698, A251699, A251700.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 07 2014