A105217 Let b(n) denote the Lucas numbers, A000032: a(n) = Sum{k=0..n}C(n,k)^2*(n-k)!*b(k).

2, 3, 11, 61, 431, 3626, 35124, 383783, 4662223, 62276683, 906637753, 14280356652, 241859495794, 4381438966659, 84512370607339, 1728802226304029, 37374059917912351, 851227845700838002, 20368894028832161532

Offset

0,1

Comments

The primes in this sequence include a(1) = 2, a(2) = 3, a(3) = 11, a(4) = 61, a(5) = 431, a(9) = 4662223. Semiprimes include a(8) = 383783 = 223 * 1721, a(16) = 1728802226304029 = 43 * 40204702937303. - Jonathan Vos Post, Apr 15 2005

If E.g.f. of b(n) is E(x) and a(n) = Sum{k = 0..n}C(n,k)^2*(n-k)!*b(k), then E.g.f. of a(n) is E(x/(1-x))/(1-x). (Thanks to Vladeta Jovovic for help.)

Links

Table of n, a(n) for n=0..18.

Formula

E.g.f. = 2*exp(x/(1-x)/2)*cosh(sqrt(5)*x/(1-x)/2)/(1-x).

Example

b(n) = 2,1,3,4,7,11,18,...
a(3) = C(3,0)^2*3!*b(0)+C(3,1)^2*2!*b(1)+C(3,2)^2*1!*b(2)+C(3,3)^2*0!*b(3) =1*6*2+9*2*1+9*1*3+1*1*4 = 12+18+27+4 = 61

Maple

b[0]:=2:b[1]:=1:for n from 2 to 30 do b[n]:=b[n-1]+b[n-2] od: > seq(sum('binomial(n, k)^2*(n-k)!*b[k]', 'k'=0..n), n=0..30);

Mathematica

Table[Sum[Binomial[n, k]^2 (n-k)!LucasL[k], {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Nov 13 2019 *)

Crossrefs

Cf. A000032.

Sequence in context: A041441 A110482 A242366 * A066046 A065597 A256394

Adjacent sequences: A105214 A105215 A105216 * A105218 A105219 A105220

Keyword

easy,nonn

Author

Miklos Kristof, Apr 13 2005

Status

approved