A228513 a(n) = Sum_{k=0..n} 2^k*(n!/k!)^2.

1, 3, 16, 152, 2448, 61232, 2204416, 108016512, 6913057024, 559957619456, 55995761946624, 6775487195543552, 975670156158275584, 164888256390748581888, 32318098252586722066432, 7271572106832012464979968, 1861522459348995191034937344, 537979990751859610209097023488

Offset: 0

Author

Vaclav Kotesovec, Aug 24 2013

Keywords

• nonn
• easy

Comments

Generally, Sum_{k=0..n} x^k*(n!/k!)^2 is asymptotic to BesselI(0,2*sqrt(x))*(n!)^2

Crossrefs

Cf. A000522, A006040, A217284.

Programs

Mathematica

Table[(n!)^2*Sum[2^j/(j!)^2, {j, 0, n}], {n, 0, 20}]
Total/@Table[2^k (n!/k!)^2,{n,0,20},{k,0,n}] (* Harvey P. Dale, Jun 10 2018 *)

Formula

a(n) = (n^2+2)*a(n-1) - 2*(n-1)^2*a(n-2).

a(n) ~ 2*Pi*BesselI(0,2*sqrt(2)) * n^(2*n+1)/exp(2*n).