A379859 - OEIS

%I #13 Jan 05 2025 09:58:57

%S 1,1,9,110,2121,53834,1720105,66197578,2984752113,154358553986,

%T 9009411908001,585917934419498,42018536835853369,3294423846094650658,

%U 280362373171289449209,25739124908062020925034,2535728977438902352557921,266836955238122741966767874,29872121613650590137264191665

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

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

%F E.g.f. A(x) satisfies A(x) = exp(-x*A(x))/(1 - x * A(x) * exp(x*A(x)))^2.

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

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*exp(x)*(1-x*exp(x))^2)/x))

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

%Y Cf. A377546, A379860.

%Y Cf. A379684.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 04 2025