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

%I #12 Mar 09 2024 08:16:06

%S 1,1,2,12,120,1320,17640,304920,6249600,143579520,3711052800,

%T 107762054400,3455138332800,120802387305600,4583177081683200,

%U 187766031131078400,8256125218115174400,387662886088250572800,19364540503274942976000,1025507260911983244595200

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

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

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

%F E.g.f.: (LambertW( -3*x^3/(1-x)^3 ) / (-3*x^3))^(1/3).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(x+exp(x^3)))/x))

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

%Y Cf. A352410, A370875.

%Y Cf. A360609.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 03 2024