A119401 a(n) = Sum_{k=0..n} (-1)^(n-k)*(n!/k!)^2*binomial(n,k).

1, 0, -3, 46, -927, 25476, -922715, 42240402, -2337537279, 147901509928, -9689806983699, 464655683171670, 44744831894861857, -27559636076854374804, 9449663596631181414933, -3046842389019074859527174, 1013788651063121586526459905

Offset: 0

Author

Vladeta Jovovic, Jul 25 2006

Keywords

• easy
• sign

Crossrefs

Cf. A119400.

Programs

Mathematica

Table[Sum[(-1)^(n - k)*(n!/k!)^2*Binomial[n, k], {k, 0, n}], {n, 0, 16}] (* Stefan Steinerberger, Jun 17 2007 *)

Formula

Sum_{n>=0} a(n)*x^n/n!^2 = BesselI(0,2*sqrt(x/(1+x)))/(1+x).

a(n) = -3*(n-1)*n*a(n-1) - 3*(n-1)^4*a(n-2) - (n-2)^3*(n-1)^3*a(n-3). - Vaclav Kotesovec, Jun 08 2019

a(n) = Sum_{k=0..n} (-1)^k*(k!)^2*binomial(n,k)^3. - Ridouane Oudra, Jul 11 2025

Extensions

More terms from Stefan Steinerberger, Jun 17 2007