A073378 Eighth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

1, 9, 63, 345, 1665, 7227, 29073, 109791, 394020, 1354210, 4486482, 14397318, 44932446, 136817370, 407566350, 1190446866, 3415935699, 9645169743, 26836557825, 73670997015, 199751003991, 535449185469

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) = A001045(n+1) see A073370 and the row polynomials of triangles A073399 and A073400.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-18,-60,234,126,-1176,36,3519,-479,-7038,144, 9408,2016,-7488,-3840,2304,2304,512).

Crossrefs

Ninth (m=8) column of triangle A073370.

Cf. A001045, A073377, A073399, A073400, A073401.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1-2*x))^9 )); // G. C. Greubel, Oct 01 2022

Mathematica

CoefficientList[Series[1/((1+x)*(1-2*x))^9, {x,0,40}], x] (* G. C. Greubel, Oct 01 2022 *)

SageMath

def A073378_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*(1-2*x))^9 ).list()
A073378_list(40) # G. C. Greubel, Oct 01 2022

Formula

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073377(k).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+8, 8) * binomial(n-k, k) * 2^k.

G.f.: 1/(1-(1+2*x)*x)^9 = 1/((1+x)*(1-2*x))^9.