A073388 Second column of the convolution triangle A073387(n+1, 1), for n >= 0.

1, 4, 16, 56, 188, 608, 1920, 5952, 18192, 54976, 164608, 489088, 1443776, 4238336, 12382208, 36022272, 104407296, 301618176, 868765696, 2495715328, 7152286720, 20452548608, 58369409024

Offset

0,2

Comments

Oribinal name: Convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself. (See the comment in A073387).

Links

Muniru A Asiru, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (4,0,-8,-4).

Formula

a(n) = Sum_{k=0..n} b(k)*b(n-k), with b(k) = A002605(k+1).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+1, 1)*binomial(n-k, k)*2^(n-k).

a(n) = ((n+1)*U(n+2) + 2*(n+2)*U(n+1))/6, with U(n) = A002605(n), n >= 0.

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

a(n) = Sum_{k=0..floor((n+2)/2)} k*binomial(n-k+2, k)2^(n-k+1). - Paul Barry, Oct 15 2004

n*a(n) +2*(-n-1)*a(n-1) +2*(-n-2)*a(n-2)=0. - R. J. Mathar, May 25 2026

Mathematica

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

Prog

(SageMath) taylor( 1/(1-2*x-2*x^2)^2, x, 0, 24).list() # Zerinvary Lajos, Jun 03 2009; modified by G. C. Greubel, Oct 03 2022
(GAP) List([0..25], n->2^n*Sum([0..Int(n/2)], k->Binomial(n-k+1, 1)*Binomial(n-k, k)*(1/2)^k)); # Muniru A Asiru, Jun 12 2018
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^2 )); // G. C. Greubel, Oct 03 2022

Crossrefs

Cf. A002605, A073387, A073388.

Sequence in context: A308288 A340257 A261386 * A109634 A026126 A026155

Adjacent sequences: A073385 A073386 A073387 * A073389 A073390 A073391

Keyword

nonn,easy,changed

Author

Wolfdieter Lang, Aug 02 2002

Extensions

New name from Wolfdieter Lang, May 06 2026

Status

approved