A389104 E.g.f. A(x) satisfies A(x) = exp( x * (1-x) * A(x) ).

1, 1, 1, -2, -31, -244, -1493, -4430, 76225, 2168920, 36418931, 450953174, 2778501409, -63700511276, -3262422613253, -90049968407654, -1793575719970687, -20042820943948240, 357444505947107683, 32821812359368723774, 1333286834019844164001, 38292697051205045089516

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Formula

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

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

Mathematica

a[0]=1; a[n_]:=n!*Sum[(-1)^(n-k)*(k+1)^(k-1)*Binomial[k, n-k]/k!, {k, 1, n}]; Table[a[n], {n, 0, 30}] (* Vincenzo Librandi, Oct 23 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n, (-1)^(n-k)*(k+1)^(k-1)*binomial(k, n-k)/k!);
(Magma) a := func< n | Factorial(n) * &+[ (-1)^(n - k) * (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, A390011.

Cf. A362771.

Sequence in context: A134179 A384479 A223145 * A188225 A164676 A381093

Adjacent sequences: A389101 A389102 A389103 * A389105 A389106 A389107

Keyword

sign

Author

Seiichi Manyama, Oct 22 2025

Status

approved