A073390 Third convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

1, 8, 48, 240, 1080, 4512, 17856, 67776, 248880, 889600, 3109376, 10664448, 35989248, 119761920, 393676800, 1280157696, 4122985728, 13165099008, 41713192960, 131243970560, 410315433984, 1275348344832

Offset: 0

Author

Wolfdieter Lang, Aug 02 2002

Keywords

• nonn
• easy

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (8,-16,-16,56,32,-64,-64,-16).

Crossrefs

Fourth (m=3) column of triangle A073387.

Cf. A002605, A073389.

Programs

GAP

List([0..25], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+3,3)*Binomial(n-k,k)*(1/2)^k)); # Muniru A Asiru, Jun 12 2018

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-2*x^2)^4 )); // G. C. Greubel, Oct 03 2022

Mathematica

CoefficientList[Series[1/(1-2*x-2*x^2)^4, {x,0,40}], x] (* G. C. Greubel, Oct 03 2022 *)

SageMath

def A073390_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2)^4 ).list()
A073390_list(40) # G. C. Greubel, Oct 03 2022

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k) with b(k) = A002605(k) and c(k) = A073389(k).

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

a(n) = ((64 + 37*n + 5*n^2)*(n+1)*U(n+1) + 4*(11 + 7*n + n^2)*(n+2)*U(n))/(6^3), with U(n) = A002605(n), n >= 0.

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