A073390 Fourth column of the convolution triangle A073387(n+3, 3), for n >= 0.

1, 8, 48, 240, 1080, 4512, 17856, 67776, 248880, 889600, 3109376, 10664448, 35989248, 119761920, 393676800, 1280157696, 4122985728, 13165099008, 41713192960, 131243970560, 410315433984, 1275348344832, 3942958252032, 12130610135040, 37151268433920

Offset

0,2

Comments

Original name: Third 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 (8,-16,-16,56,32,-64,-64,-16).

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k) with b(k) = A002605(k+1) 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+2) + 4*(11 + 7*n + n^2)*(n+2)*U(n+1))/(6^3), with U(n) = A002605(n), n >= 0.

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

Mathematica

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

Prog

(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<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-2*x^2)^4 )); // G. C. Greubel, Oct 03 2022
(SageMath)
def A073390_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2)^4 ).list()
A073390_list(40) # G. C. Greubel, Oct 03 2022

Crossrefs

Cf. A002605, A073387. A073389.

Sequence in context: A211012 A081084 A230931 * A069021 A079763 A079785

Adjacent sequences: A073387 A073388 A073389 * A073391 A073392 A073393

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 02 2002

Extensions

New name from Wolfdieter Lang, May 06 2026

Status

approved