A073392 Fifth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

1, 12, 96, 616, 3444, 17472, 82432, 367488, 1565280, 6421376, 25525248, 98773248, 373450112, 1383674880, 5036089344, 18041821184, 63727070976, 222249968640, 766234140672, 2614196680704, 8834194123776

Offset: 0

Author

Wolfdieter Lang, Aug 02 2002

Keywords

• nonn
• easy

Examples

			x^6 + 12*x^7 + 96*x^8 + 616*x^9 + 3444*x^10 + ... + 222249968640*x^23 + 766234140672*x^24 + 2614196680704*x^25 + 8834194123776*x^26 + ... - _Zerinvary Lajos_, Jun 03 2009
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (12,-48,40,180,-288,-384,576,720,-320,-768,-384,-64).

Crossrefs

Sixth (m=5) column of triangle A073387.

Cf. A002605, A073391.

Programs

GAP

List([0..30], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+5,5)*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)^6 )); // G. C. Greubel, Oct 04 2022

Mathematica

CoefficientList[Series[1/(1-2*x*(1+x))^6, {x,0,30}],x] (* Harvey P. Dale, May 12 2018 *)

Sage

taylor( 1/(1-2*x-2*x^2)^6, x, 0, 30).list() # Zerinvary Lajos, Jun 03 2009; modified by G. C. Greubel, Oct 04 2022

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A002605(k) and c(k) = A073391(k).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+5, 5)*binomial(n-k, k)*2^(n-k).

a(n) = (n+4)*(n+8)*((19*n^2 + 158*n + 275)*(n+1)*U(n+1) + 2*(7*n^2 + 52*n + 65)*(n+2)*U(n))/(2^6*3^4*5), with U(n) = A002605(n), n >= 0.

G.f.: 1/(1-2*x*(1+x))^6.