A377398 Expansion of e.g.f. (2 - exp(x))^3.

1, -3, 3, 9, 3, -63, -357, -1431, -5037, -16623, -52917, -164871, -506877, -1545183, -4684677, -14152311, -42653517, -128353743, -385847637, -1159115751, -3480492957, -10447770303, -31355893797, -94092847191, -282328873197, -847087282863, -2541463175157

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Formula

a(n) = 5*a(n-1) - 6*a(n-2) - 24 for n > 2.

a(n) = Sum_{k=0..3} (-1)^k * k! * binomial(3,k) * Stirling2(n,k).

a(n) = Sum_{k=0..3} (-1)^k * 2^(3-k) * binomial(3,k) * k^n.

G.f.: (1-4*x) * (1-5*x+12*x^2)/((1-x) * (1-2*x) * (1-3*x)).

a(n) = 3*2^(n+1) - 3^n - 12 for n > 0. - Stefano Spezia, Oct 27 2024

a(0) = 1; a(n) = Sum_{k=1..n} (1 - 4 * k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Oct 27 2024

Prog

(PARI) a(n) = sum(k=0, 3, (-1)^k*k!*binomial(3, k)*stirling(n, k, 2));
(PARI) a(n) = sum(k=0, 3, (-1)^k*2^(3-k)*binomial(3, k)*k^n);

Crossrefs

Cf. A226515, A377399.

Sequence in context: A113213 A088032 A348397 * A066572 A307379 A276147

Adjacent sequences: A377395 A377396 A377397 * A377399 A377400 A377401

Keyword

sign,easy

Author

Seiichi Manyama, Oct 27 2024

Status

approved