A373392 Inverse binomial transform of A135318.

1, 0, 0, 1, -2, 3, -4, 7, -18, 51, -136, 339, -814, 1935, -4620, 11111, -26842, 64923, -156944, 379067, -915078, 2208711, -5331476, 12870639, -31072754, 75018195, -181113240, 437248771, -1055610782, 2548462143, -6152518684, 14853483127, -35859484938, 86572485771

Offset: 0

Author

Paul Curtz, Jun 03 2024

Keywords

• sign
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (-4,-5,-2,2).

Crossrefs

Cf. A000129, A009545, A135318, A373358.

Programs

Mathematica

LinearRecurrence[{-4, -5, -2, 2}, {1, 0, 0, 1}, 35] (* Amiram Eldar, Jun 09 2024 *)

PARI

a(n) = ((-([-2,-1;-1, 0]^(n-2))[2, 1]) - 2*((I-1)^(n-4) + (-I-1)^(n-4)))/3; \\ Thomas Scheuerle, Jun 04 2024

Formula

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

E.g.f.: 1/6*exp(-x)*(2*cos(-x) + 4*cosh(sqrt(2)*-x) - 3*sqrt(2)*sinh(sqrt(2)*-x)).

a(n) = -4*a(n-1) - 5*a(n-2) - 2*a(n-3) + 2*a(n-4), for n > 4.

a(n) = (-1)^(n+1)*(A000129(n-2) + 2*A009545(n-2))/3, for n > 2. - Thomas Scheuerle, Jun 04 2024

a(n) = A373358(n-3) - (-1)^n*A009545(n+2) for n > 2.