A073380 Third convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

1, 8, 44, 200, 810, 3032, 10716, 36248, 118435, 376240, 1167720, 3553840, 10636180, 31375440, 91392040, 263266512, 750922021, 2123059448, 5955034740, 16584106040, 45884989054, 126202397032

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 (8,-20,8,26,-8,-20,-8,-1).

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A054457(k).

a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n-k+3, 3) * binomial(n-k, k).

a(n) = ((147 +94*n +14*n^2)*(n+1)*U(n+1) + 3*(15 +12*n +2*n^2)*(n+2)*U(n))/ (3*2^7), with U(n) = A000129(n+1), n >= 0.

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

a(n) = F'''(n+4, 2)/6, that is, 1/6 times the 3rd derivative of the (n+4)th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006

Mathematica

CoefficientList[Series[1/(1-2*x-x^2)^4, {x, 0, 40}], x] (* G. C. Greubel, Oct 02 2022 *)
LinearRecurrence[{8, -20, 8, 26, -8, -20, -8, -1}, {1, 8, 44, 200, 810, 3032, 10716, 36248}, 30] (* Harvey P. Dale, Feb 18 2023 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^4 )); // G. C. Greubel, Oct 02 2022
(SageMath)
def A073380_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^2)^4 ).list()
A073380_list(30) # G. C. Greubel, Oct 02 2022

Crossrefs

Fourth (m=3) column of triangle A054456, A054457 (m=2).

Cf. A000129.

Sequence in context: A283077 A023007 A169795 * A273603 A270902 A241395

Adjacent sequences: A073377 A073378 A073379 * A073381 A073382 A073383

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 02 2002

Status

approved