A073381 Fourth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

1, 10, 65, 340, 1555, 6482, 25235, 93200, 330070, 1129580, 3756950, 12197320, 38787770, 121148300, 372476410, 1129367632, 3382133695, 10016694470, 29370557375, 85341915260, 245939376949, 703423066190

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Milan Janjic, Hessenberg Matrices and Integer Sequences , J. Int. Seq. 13 (2010) # 10.7.8.

Index entries for linear recurrences with constant coefficients, signature (10,-35,40,30,-68,-30,40,35,10,1).

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A073380(k).

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

a(n) = ((2457 +2128*n +572*n^2 +48*n^3)*(n+1)*U(n+1) + 5*(123 +142*n +44*n^2 +4*n^3) *(n+2)*U(n))/(3*2^11), with U(n) = A000129(n+1), n>=0.

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

a(n) = F''''(n+5, 2)/4!, that is, 1/4! times the 4th derivative of the (n+5)th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006

Mathematica

CoefficientList[Series[1/(1-2*x-x^2)^5, {x, 0, 40}], x] (* G. C. Greubel, Oct 02 2022 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^5 )); // G. C. Greubel, Oct 02 2022
(SageMath)
def A073381_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^2)^5 ).list()
A073381_list(40) # G. C. Greubel, Oct 02 2022

Crossrefs

Fifth (m=4) column of triangle A054456, A073380.

Cf. A000129.

Sequence in context: A160458 A023009 A169797 * A092441 A022638 A354393

Adjacent sequences: A073378 A073379 A073380 * A073382 A073383 A073384

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 02 2002

Status

approved