A390856 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k+1,n-k) * Fibonacci(k+1).

1, 1, 0, 2, 14, 40, 85, 194, 531, 1488, 3933, 10078, 26068, 68503, 180428, 472781, 1235331, 3230628, 8460734, 22162520, 58026826, 151889301, 397612420, 1040994327, 2725491954, 7135481535, 18680634840, 48906105174, 128038194911, 335209825828, 877591282573

Offset

0,4

Links

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

Index entries for linear recurrences with constant coefficients, signature (5,-11,17,-14,6,-1).

Formula

G.f.: 1/((1-x)^2 * (1+g-g^2)), where g = x/(1-x)^3.

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

a(n) = 5*a(n-1) - 11*a(n-2) + 17*a(n-3) - 14*a(n-4) + 6*a(n-5) - a(n-6).

Mathematica

Table[Sum[(-1)^k*Binomial[n+2*k+1, n-k]*Fibonacci[k+1], {k, 0, n}], {n, 0, 40}] (* Vincenzo Librandi, Nov 26 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+2*k+1, n-k)*fibonacci(k+1));
(Magma) [&+[(-1)^k*Binomial(n+2*k+1, n-k)*Fibonacci(k+1): k in [0..n]] : n in [0..40] ]; // Vincenzo Librandi, Nov 26 2025

Crossrefs

Partial sums of A390855.

Cf. A000045, A000217, A390826.

Sequence in context: A216528 A329735 A290124 * A192349 A230801 A231081

Adjacent sequences: A390853 A390854 A390855 * A390857 A390858 A390859

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 21 2025

Status

approved