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Expansion of e.g.f. (1/x) * Series_Reversion( 4*x/(3 + exp(4*x)) ).
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%I #11 Nov 07 2024 15:41:51

%S 1,1,6,58,824,15576,368560,10494352,349680000,13354956160,

%T 575343613184,27606884967168,1460295317318656,84429863673895936,

%U 5297505756426098688,358520710389920598016,26033795963713021116416,2019060825791610516504576

%N Expansion of e.g.f. (1/x) * Series_Reversion( 4*x/(3 + exp(4*x)) ).

%H <a href="/index/Res#revert">Index entries for reversions of series</a>

%F a(n) = 1/(4*(n+1)) * Sum_{k=0..n+1} 3^(n+1-k) * k^n * binomial(n+1,k).

%F a(n) = n! * Sum_{k=0..n} 4^(n-k) * Stirling2(n,k)/(n-k+1)!. - _Seiichi Manyama_, Nov 07 2024

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(4*x/(3+exp(4*x)))/x))

%o (PARI) a(n) = sum(k=0, n+1, 3^(n+1-k)*k^n*binomial(n+1, k))/(4*(n+1));

%Y Cf. A201595, A370907.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 05 2024