A060931 Fourth convolution of Lucas numbers A000032(n+1), n >= 0.

1, 15, 110, 545, 2120, 7043, 20965, 57560, 148545, 365045, 862224, 1970905, 4382820, 9520315, 20265665, 42385132, 87284120, 177293730, 355738710, 705980760, 1387213926, 2701362950, 5217448800, 10001654350

Offset: 0

Author

Wolfdieter Lang, Apr 20 2001

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-5,-10,15,11,-15,-10,5,5,1).

Crossrefs

Cf. A000032, A000204, A004799, A060922, A060929, A060930.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( ((1+2*x)/(1-x-x^2))^5 )); // G. C. Greubel, Apr 08 2021

Mathematica

Table[((n+1)/120)*((5*n^3+5*n^2-10*n+72)*LucasL[n+5] + 4*(5*n^2+10*n-24)*LucasL[n+ 4]), {n, 0, 40}] (* G. C. Greubel, Apr 08 2021 *)

Sage

def A060931_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1+2*x)/(1-x-x^2))^5 ).list()
A060931_list(40) # G. C. Greubel, Apr 08 2021

Formula

a(n) = A060921(n+4, 4) (fifth column of Lucas triangle).

a(n) = (n+1)*( (15*n^3 +55*n^2 +50*n +24)*L(n+2) + 2*(5*n^3 +15*n^2 +10*n +24)*L(n+1))/5!, with the Lucas numbers L(n)=A000032(n).

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