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Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - log(1+2*x)/2) ).
1

%I #11 Mar 06 2024 11:49:10

%S 1,1,2,8,48,384,3872,47088,671360,10985088,202927872,4178030592,

%T 94874787840,2355758714880,63498696376320,1846607063998464,

%U 57630620308930560,1921296165774950400,68145277700464312320,2562234152415762972672,101801592691389968154624

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

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

%F a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^(n-k) * (n+k)! * Stirling1(n,k).

%F a(n) ~ 2^(2*n + 1) * LambertW(exp(-1))^n * n^(n-1) / (sqrt(1 + LambertW(exp(-1))) * exp(n) * (1 - LambertW(exp(-1)))^(2*n + 1)). - _Vaclav Kotesovec_, Mar 06 2024

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-log(1+2*x)/2))/x))

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

%Y Cf. A198860, A370937.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 06 2024