A377646 Expansion of e.g.f. (1 + x * (exp(x) - 1))^2.

1, 0, 4, 6, 32, 130, 432, 1274, 3488, 9090, 22880, 56122, 134928, 319202, 745136, 1719930, 3931712, 8912386, 20053440, 44825978, 99614000, 220200162, 484441232, 1061157946, 2315254752, 5033163650, 10905189152, 23555209914, 50734299728, 108984793570

Offset

0,3

Links

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

Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

Formula

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

From Andrew Howroyd, Nov 13 2025: (Start)

a(n) = n*(n - 1)*2^(n - 2) - 2*n*(n - 2) for n >= 3.

G.f.: (1 - 9*x + 37*x^2 - 93*x^3 + 176*x^4 - 248*x^5 + 212*x^6 - 88*x^7 + 16*x^8)/((1 - x)^3*(1 - 2*x)^3). (End)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, k!*binomial(2, k)*stirling(n-k, k, 2)/(n-k)!);
(PARI) a(n) = if(n < 3, [1, 0, 4][n+1], n*(n-1)*2^(n-2) - 2*n*(n-2)); \\ Andrew Howroyd, Nov 13 2025

Crossrefs

Cf. A377680, A377681.

Cf. A375660.

Sequence in context: A239224 A087299 A229712 * A331513 A164127 A180139

Adjacent sequences: A377643 A377644 A377645 * A377647 A377648 A377649

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 04 2024

Status

approved