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A251695 a(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n.

%I A251695 #10 Sep 08 2022 08:46:10
%S A251695 1,4,175,16250,2348125,463050000,115966796875,35253537343750,
%T A251695 12611991884765625,5191587030710937500,2417311348659677734375,
%U A251695 1256208098030090332031250,720779749270420907470703125,452589644988876542822265625000,308707218248583408960223388671875
%N A251695 a(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n.
%H A251695 G. C. Greubel, <a href="/A251695/b251695.txt">Table of n, a(n) for n = 0..268</a>
%F A251695 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 A251695 (1) A(x) = exp( 5*x*A(x)^4 * G(x*A(x)^4)^4 ) / G(x*A(x)^4).
%F A251695 (2) A(x) = F(x*A(x)^4) where F(x) = exp(5*x*G(x)^4)/G(x) is the e.g.f. of A251665.
%F A251695 (3) A(x) = ( Series_Reversion( x*G(x)^4 / exp(20*x*G(x)^4) )/x )^(1/4).
%F A251695 E.g.f.: (-LambertW(-20*x)/(20*x))^(1/4) * (1 + LambertW(-20*x)/20). - _Vaclav Kotesovec_, Dec 07 2014
%e A251695 E.g.f.: A(x) = 1 + 4*x + 175*x^2/2! + 16250*x^3/3! + 2348125*x^4/4! + 463050000*x^5/5! +...
%e A251695 such that A(x) = exp( 5*x*A(x)^4 * G(x*A(x)^4)^4 ) / G(x*A(x)^4),
%e A251695 where G(x) = 1 + x*G(x)^5 is the g.f. A002294:
%e A251695 G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 +...
%e A251695 Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^4) where
%e A251695 F(x) = 1 + 4*x + 47*x^2/2! + 1034*x^3/3! + 34349*x^4/4! + 1540480*x^5/5! +...
%e A251695 F(x) = exp( 5*x*G(x)^4 ) / G(x) is the e.g.f. of A251665.
%t A251695 Table[(3*n + 1)*(4*n + 1)^(n - 2)*5^n, {n, 0, 50}] (* _G. C. Greubel_, Nov 13 2017 *)
%o A251695 (PARI) {a(n) = (3*n+1) * (4*n+1)^(n-2) * 5^n}
%o A251695 for(n=0,20,print1(a(n),", "))
%o A251695 (PARI) {a(n)=local(G=1,A=1); for(i=0, n, G = 1 + x*G^5 +x*O(x^n));
%o A251695 A = ( serreverse( x*G^4 / exp(20*x*G^4) )/x )^(1/4); n!*polcoeff(A, n)}
%o A251695 for(n=0, 20, print1(a(n), ", "))
%o A251695 (Magma) [(3*n + 1)*(4*n + 1)^(n - 2)*5^n: n in [0..50]]; // _G. C. Greubel_, Nov 13 2017
%Y A251695 Cf. A251665, A002294.
%Y A251695 Cf. Variants: A127670, A251693, A251694, A251696, A251697, A251698, A251699, A251700.
%K A251695 nonn
%O A251695 0,2
%A A251695 _Paul D. Hanna_, Dec 07 2014