A060933 Sixth convolution of Lucas numbers A000032(n+1), n >= 0.

1, 21, 217, 1498, 7910, 34566, 131446, 449732, 1416513, 4174765, 11651717, 31075422, 79751854, 198036146, 477899790, 1124785648, 2589534248, 5845989156, 12968091584, 28316428700, 60953528230, 129515454530, 271955244610, 564879359940, 1161646929275, 2366938010983, 4781794056543

Offset: 0

Author

Wolfdieter Lang, Apr 20 2001

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-14,-7,49,-14,-77,29,77,-14,-49,-7,14,7,1).

Crossrefs

Cf. A000032, A000204, A004799, A060922, A060929, A060930, A060931, A060932.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( ((1+2*x)/(1-x-x^2))^7 )); // G. C. Greubel, Apr 08 2021

Maple

m:= 40; S:= series( ((1+2*x)/(1-x-x^2))^7, x, m+1);
seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Apr 08 2021

Mathematica

Table[(n+1)(2(100n^5+845n^4+2480n^3+4345n^2+5910n+2952)LucasL[n+2]+(125n^5+ 1030n^4+2995n^3+5930n^2+8280n+288)LucasL[n+1])/18000,{n,0,30}] (* Harvey P. Dale, Aug 13 2013 *)
CoefficientList[Series[((1+2x)/(1-x-x^2))^7, {x,0,30}], x] (* Vincenzo Librandi, Aug 13 2013 *)

Sage

def A060930_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( ((1+2*x)/(1-x-x^2))^7 ).list()
A060930_list(40) # G. C. Greubel, Apr 08 2021

Formula

G.f.: ( (1+2*x) / (1-x-x^2) )^7.

a(n) = A060922(n+6, 6) (seventh column of Lucas triangle).

a(n) = (n+1)*(2*(100*n^5 +845*n^4 +2480*n^3 +4345*n^2 +5910*n +2952)*L(n+2) + (125*n^5 +1030*n^4 +2995*n^3 +5930*n^2 +8280*n +288)*L(n+1))/(6!*5^2), with the Lucas numbers L(n)=A000032(n).