A073391 Fifth column of the convolution triangle A073387(n+4 4), for n >= 0.

1, 10, 70, 400, 2020, 9352, 40600, 167680, 665440, 2555840, 9551936, 34880000, 124853120, 439228160, 1521839360, 5202292736, 17571249920, 58712184320, 194280061440, 637228462080, 2073332481024, 6696470231040

Offset

0,2

Comments

Original name: Fourth 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 (10,-30,0,120,-48,-240,0,240,160,32).

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A002605(k+1) and c(k) = A073390(k).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+4, 4)*binomial(n-k, k)*2^(n-k).

a(n) = (2*(419 + 326*n + 79*n^2 + 6*n^3)*(n+1)*U(n+2) + (458 + 421*n + 112*n^2 + 9*n^3)*(n+2)*U(n+1))/(2^5*3^4), with U(n) = A002605(n), n >= 0.

G.f.: 1/(1-2*x*(1+x))^5.

Mathematica

CoefficientList[Series[1/(1-2*x-2*x^2)^5, {x, 0, 40}], x] (* G. C. Greubel, Oct 04 2022 *)

Prog

(GAP) List([0..25], n->2^n*Sum([0..Int(n/2)], k->Binomial(n-k+4, 4)*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)^5 )); // G. C. Greubel, Oct 04 2022
(SageMath)
def A073391_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2)^5 ).list()
A073391_list(40) # G. C. Greubel, Oct 04 2022

Crossrefs

Cf. A002605, A073387, A073390.

Sequence in context: A271040 A005567 A174434 * A002802 A101029 A375921

Adjacent sequences: A073388 A073389 A073390 * A073392 A073393 A073394

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 02 2002

Extensions

New name from Wolfdieter Lang, May 06 2026

Status

approved