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A381988 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

%I A381988 #17 Mar 14 2025 09:00:43
%S A381988 1,2,15,313,10773,510981,30876463,2267990159,196204786025,
%T A381988 19539828320905,2201822913234771,276969947671828995,
%U A381988 38473403439454795837,5849221857618942870029,966078641687956464576119,172251173569831561500070711,32975613823747758363130520529,6746227557293225645352382744593
%N A381988 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.
%F A381988 Let F(x) be the e.g.f. of A377526. F(x) = B(x*A(x)) = exp( 1/4 * Sum_{k>=1} binomial(4*k,k) * (x*A(x))^k/k ).
%F A381988 a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * A002294(k)/(n-k)!.
%o A381988 (PARI) a(n) = n!*sum(k=0, n, (k+1)^(n-k)*binomial(5*k+1, k)/((5*k+1)*(n-k)!));
%Y A381988 Cf. A381987, A381989.
%Y A381988 Cf. A002293, A002294, A346647, A377526.
%K A381988 nonn
%O A381988 0,2
%A A381988 _Seiichi Manyama_, Mar 12 2025