A390851 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k,n-k) * Fibonacci(k+1).

1, 0, 0, 2, 5, 9, 18, 40, 88, 189, 405, 872, 1880, 4050, 8721, 18781, 40450, 87120, 187632, 404105, 870329, 1874448, 4037040, 8694658, 18725869, 40330305, 86860242, 187072760, 402902600, 867739941, 1868870061, 4025025400, 8668783240, 18670143762, 40210287705, 86601756149, 186516053378, 401703611040

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-1).

Formula

G.f.: 1/((1-x) * (1+g-g^2)), where g = x/(1-x)^2.

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

a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3) - a(n-4).

Mathematica

Table[Sum[(-1)^k*Binomial[n+k, n-k]*Fibonacci[k+1], {k, 0, n}], {n, 0, 40}] (* Vincenzo Librandi, Nov 24 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+k, n-k)*fibonacci(k+1));
(Magma) [&+[(-1)^k*Binomial(n+k, n-k)*Fibonacci(k+1): k in [0..n]] : n in [0..40] ]; // Vincenzo Librandi, Nov 24 2025

Crossrefs

Partial sums of A390850.

Cf. A000045, A389699.

Sequence in context: A293354 A293329 A152546 * A286713 A342013 A213544

Adjacent sequences: A390848 A390849 A390850 * A390852 A390853 A390854

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 21 2025

Status

approved