A073386 Ninth convolution of A000129(n+1) (generalized (2,1)-Fibonacci, called Pell numbers), n>=0, with itself.

1, 20, 230, 1980, 14135, 88264, 497860, 2591160, 12630475, 58295380, 256887774, 1087825180, 4449607565, 17654254880, 68177369040, 257006941664, 948023601910, 3428968838680, 12182953719860

Offset: 0

Author

Wolfdieter Lang, Aug 02 2002

Keywords

• nonn
• easy

Comments

For a(n) in terms of U(n+1) and U(n) with U(n) = A000129(n+1) see the row polynomials of triangles A058402 and A058403 and the comment there.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (20,-170,780,-1965,2064,1800,-6480,1710,8600,-3772, -8600,1710,6480,1800,-2064,-1965,-780,-170,-20,-1).

Crossrefs

Tenth (m=9) column of triangle A054456.

Cf. A000129, A058402, A058403, A073385.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-x^2)^10 )); // G. C. Greubel, Oct 03 2022

Mathematica

CoefficientList[Series[1/(1-2*x-x^2)^10, {x,0,40}], x] (* G. C. Greubel, Oct 03 2022 *)
LinearRecurrence[{20,-170,780,-1965,2064,1800,-6480,1710,8600,-3772,-8600,1710,6480,1800,-2064,-1965,-780,-170,-20,-1},{1,20,230,1980,14135,88264,497860,2591160,12630475,58295380,256887774,1087825180,4449607565,17654254880,68177369040,257006941664,948023601910,3428968838680,12182953719860,42585118702280},20] (* Harvey P. Dale, Nov 20 2022 *)

SageMath

def A073386_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-x^2)^10 ).list()
A073386_list(40) # G. C. Greubel, Oct 03 2022

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A000129(k+1) and c(k) = A073385(k).

a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k)*binomial(n-k+9, 9)*binomial(n-k, k).

G.f.: 1/(1-(2+x)*x)^10.

a(n) = F'''''''''(n+10, 2)/9!, that is, 1/9! times the 9th derivative of the (n+10)th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006