A377680 Expansion of e.g.f. (1 + x * (exp(x) - 1))^3.

1, 0, 6, 9, 84, 375, 1998, 11361, 60840, 299403, 1368930, 5906373, 24362748, 97019247, 375712470, 1422455625, 5286155088, 19340722707, 69831127242, 249265052301, 880927979940, 3086000399223, 10726216043070, 37020328044945, 126961071656184, 432900077950875

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (24,-260,1680,-7206,21600,-46364,71760,-79441,61320,-31320,9504,-1296).

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} k! * binomial(3,k) * Stirling2(n-k,k)/(n-k)!.

G.f.: (1 - 24*x + 266*x^2 - 1815*x^3 + 8634*x^4 - 30981*x^5 + 89318*x^6 - 216717*x^7 + 446185*x^8 - 750291*x^9 + 969468*x^10 - 907596*x^11 + 585648*x^12 - 248832*x^13 + 64800*x^14 - 7776*x^15)/((1 - x)*(1 - 2*x)*(1 - 3*x))^4. - Andrew Howroyd, Nov 13 2025

Prog

(PARI) a(n) = n!*sum(k=0, min(3, n\2), k!*binomial(3, k)*stirling(n-k, k, 2)/(n-k)!);

Crossrefs

Cf. A377646, A377681.

Cf. A375661.

Sequence in context: A204383 A266006 A372412 * A103107 A370623 A377683

Adjacent sequences: A377677 A377678 A377679 * A377681 A377682 A377683

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 04 2024

Status

approved