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

Author

Wolfdieter Lang, Aug 02 2002

Keywords

• nonn
• easy

Links

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

Crossrefs

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

Cf. A000129.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^5 )); // G. C. Greubel, Oct 02 2022

Mathematica

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

SageMath

def A073381_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^2)^5 ).list()
A073381_list(40) # G. C. Greubel, Oct 02 2022

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