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A060930
Third convolution of Lucas numbers A000032(n+1), n >= 0.
4
1, 12, 70, 280, 905, 2568, 6666, 16220, 37580, 83780, 181074, 381488, 786715, 1593160, 3176210, 6246732, 12139859, 23344760, 44471340, 84005640, 157483176, 293201912, 542468100, 997906400, 1826073525
OFFSET
0,2
FORMULA
G.f.: ((1+2*x)/(1-x-x^2))^4.
a(n) = A060922(n+3, 3) (fourth column of Lucas triangle).
a(n) = (2*(25*n^3 + 60*n^2 + 35*n +24)*L(n+2) + (25*n^3 + 90*n^2 + 95*n + 6)*L(n+1))/(3!*5^2), with the Lucas numbers L(n) = A000032(n).
MATHEMATICA
Table[((25*n^3+90*n^2+95*n+6)*LucasL[n+4] -12*(5*n^2+10*n-3)*LucasL[n+2])/150, {n, 0, 40}] (* G. C. Greubel, Apr 08 2021 *)
PROG
(Magma)
R<x>:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( ((1+2*x)/(1-x-x^2))^4 )); // G. C. Greubel, Apr 08 2021
(Sage)
def A060930_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( ((1+2*x)/(1-x-x^2))^4 ).list()
A060930_list(40) # G. C. Greubel, Apr 08 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Apr 20 2001
STATUS
approved