OFFSET
0,3
FORMULA
Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) * (Ei(x) - log(x) - gamma).
From Vaclav Kotesovec, Mar 06 2022: (Start)
Recurrence: n*(n^3 - 12*n^2 + 46*n - 50)*a(n) = (2*n^5 - 24*n^4 + 95*n^3 - 123*n^2 + 37*n - 2)*a(n-1) - (n^6 - 12*n^5 + 51*n^4 - 95*n^3 + 121*n^2 - 139*n + 74)*a(n-2) + (n-2)*(2*n^5 - 27*n^4 + 140*n^3 - 344*n^2 + 409*n - 173)*a(n-3) - (n-3)^2*(n-2)*(n^3 - 9*n^2 + 25*n - 15)*a(n-4).
a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^(n - 3/4) / sqrt(2). (End)
MATHEMATICA
Table[Sum[Binomial[n, k]^2 (n - k - 1)!, {k, 0, n - 1}], {n, 0, 22}]
nmax = 22; Assuming[x > 0, CoefficientList[Series[BesselI[0, 2 Sqrt[x]] (ExpIntegralEi[x] - Log[x] - EulerGamma), {x, 0, nmax}], x]] Range[0, nmax]!^2
PROG
(PARI) a(n) = sum(k=0, n-1, binomial(n, k)^2 * (n-k-1)!); \\ Michel Marcus, Mar 06 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 06 2022
STATUS
approved