A352149 a(n) = Sum_{k=0..n-1} binomial(n,k)^2 * (n-k-1)!.

0, 1, 5, 20, 90, 499, 3395, 27474, 256984, 2720169, 32080501, 416574212, 5900292266, 90461885331, 1491788697451, 26318520300986, 494449968500832, 9852544385880961, 207497251731808341, 4604297325494524516, 107348917822006139114, 2623224641748615607715, 67035139167875735937219

Offset

0,3

Links

Table of n, a(n) for n=0..22.

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

Cf. A002104, A002720, A352150.

Sequence in context: A360076 A365115 A375455 * A196532 A377152 A002745

Adjacent sequences: A352146 A352147 A352148 * A352150 A352151 A352152

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 06 2022

Status

approved