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

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

Crossrefs

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

Cf. A000129.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^4 )); // G. C. Greubel, Oct 02 2022

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 *)

SageMath

def A073380_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^2)^4 ).list()
A073380_list(30) # 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) = 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