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Expansion of e.g.f. (1/x) * Series_Reversion( 3*x/(2 + exp(3*x)) ).
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%I #13 Nov 07 2024 15:41:47

%S 1,1,5,42,519,8526,175329,4338594,125632035,4169652390,156101072373,

%T 6508965708378,299190004799679,15031796956994286,819581031710623017,

%U 48199003176462356754,3041324249730311069595,204962505644116505863926

%N Expansion of e.g.f. (1/x) * Series_Reversion( 3*x/(2 + exp(3*x)) ).

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

%F a(n) = 1/(3*(n+1)) * Sum_{k=0..n+1} 2^(n+1-k) * k^n * binomial(n+1,k).

%F a(n) = n! * Sum_{k=0..n} 3^(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(3*x/(2+exp(3*x)))/x))

%o (PARI) a(n) = sum(k=0, n+1, 2^(n+1-k)*k^n*binomial(n+1, k))/(3*(n+1));

%Y Cf. A201595, A370908.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 05 2024