A377399 Expansion of e.g.f. (2 - exp(x))^4.

1, -4, 8, 8, -40, -184, -232, 1928, 19160, 116936, 600728, 2826248, 12623960, 54550856, 230564888, 959736968, 3952166360, 16149893576, 65626404248, 265592398088, 1071642518360, 4314414017096, 17341238230808, 69615800073608, 279215943071960, 1119122403273416

Offset: 0

Author

Seiichi Manyama, Oct 27 2024

Keywords

• sign
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Crossrefs

Cf. A226738.

Programs

PARI

a(n) = sum(k=0, 4, (-1)^k*k!*binomial(4, k)*stirling(n, k, 2));

PARI

a(n) = sum(k=0, 4, (-1)^k*2^(4-k)*binomial(4, k)*k^n);

Formula

a(n) = 9*a(n-1) - 26*a(n-2) + 24*a(n-3) + 192 for n > 3.

a(n) = Sum_{k=0..4} (-1)^k * k! * binomial(4,k) * Stirling2(n,k).

a(n) = Sum_{k=0..4} (-1)^k * 2^(4-k) * binomial(4,k) * k^n.

G.f.: (1-14*x+83*x^2-262*x^3+384*x^4)/((1-x) * (1-2*x) * (1-3*x) * (1-4*x)).

a(n) = 4^n - 8*3^n + 3*2^(n+3) - 32 for n > 0. - Stefano Spezia, Oct 27 2024

a(0) = 1; a(n) = Sum_{k=1..n} (1 - 5 * k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Oct 27 2024