A073391 Fourth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

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

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 (10,-30,0,120,-48,-240,0,240,160,32).

Crossrefs

Fifth (m=4) column of triangle A073387.

Cf. A002605, A073390.

Programs

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:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-2*x^2)^5 )); // G. C. Greubel, Oct 04 2022

Mathematica

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

SageMath

def A073391_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2)^5 ).list()
A073391_list(40) # 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) = 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+1) + (458 + 421*n + 112*n^2 + 9*n^3)*(n+2)*U(n))/(2^5*3^4), with U(n) = A002605(n), n >= 0.

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