A073389 Second convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

1, 6, 30, 128, 504, 1872, 6672, 23040, 77616, 256288, 832416, 2666496, 8441600, 26454528, 82174464, 253280256, 775316736, 2358812160, 7137023488, 21487386624, 64401106944, 192229535744, 571630694400, 1693996941312, 5004131659776, 14738997288960, 43293528760320

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 (6,-6,-16,12,24,8).

Crossrefs

Third (m=2) column of triangle A073387, A073388.

Cf. A002605.

Programs

GAP

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

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^3 )); // G. C. Greubel, Oct 03 2022

Mathematica

CoefficientList[Series[1/(1-2x(1+x))^3,{x,0,25}],x]  (* Harvey P. Dale, Mar 14 2011 *)

Sage

taylor( 1/(1-2*x-2*x^2)^3, x, 0, 25).list() # Zerinvary Lajos, Jun 03 2009; modified by 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) = A073388(k).

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

a(n) = (n+3)*((n+1)*U(n+1) + (n+2)*U(n))/12, with U(n) = A002605(n), n >= 0.

G.f.: 1/(1-2*x*(1+x))^3.