A390011 E.g.f. A(x) satisfies A(x) = exp( x * (1-x) * A(x)^2 ).

1, 1, 3, 19, 201, 3001, 57403, 1333179, 36380753, 1140542065, 40396085811, 1595414047459, 69523106792857, 3313793538584361, 171510379926687467, 9579129246449905291, 574261910848799455137, 36780943846496265972961, 2506644104953786816077667

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..360

Formula

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

E.g.f.: exp( -LambertW(-2*x * (1-x))/2 ).

a(n) ~ sqrt(2*exp(-1) + sqrt(1 - 2*exp(-1)) - 1) * 2^(n-1) * n^(n-1) / (exp(n-1) * (1 - sqrt(1 - 2*exp(-1)))^n). - Vaclav Kotesovec, Oct 23 2025

Mathematica

a[n_]:=n!*Sum[(-1)^(n-k)*(2*k+1)^(k-1)*Binomial[k, n-k]/k!, {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Vincenzo Librandi, Oct 23 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n, (-1)^(n-k)*(2*k+1)^(k-1)*binomial(k, n-k)/k!);
(Magma) a := func< n | Factorial(n) * &+[ (-1)^(n - k) * (2*k + 1)^(k - 1) * Binomial(k, n - k) / Factorial(k) : k in [0..n] ] >;
[ a(n) : n in [0..25] ]; // Vincenzo Librandi, Oct 23 2025

Crossrefs

Cf. A293604, A389104.

Cf. A362773, A365039.

Sequence in context: A288693 A195895 A127502 * A175176 A362968 A233336

Adjacent sequences: A390008 A390009 A390010 * A390012 A390013 A390014

Keyword

nonn

Author

Seiichi Manyama, Oct 22 2025

Status

approved