A294573 a(n) = n! * [x^n] exp((n+1)*x)*BesselI(1,2*x)/x.

1, 2, 10, 76, 777, 9996, 155139, 2821400, 58856963, 1385621260, 36343079188, 1051024082472, 33226817252215, 1140040324751160, 42193259673938754, 1675570154136359472, 71069261432474378715, 3206616936773061141900, 153358034674756782660342, 7749560706936442485607560

Offset: 0

Author

Ilya Gutkovskiy, Nov 02 2017

Keywords

• nonn

Comments

The n-th term of the n-th binomial transform of A001006.

Links

• Robert Israel, Table of n, a(n) for n = 0..300
• N. J. A. Sloane, Transforms

Crossrefs

Diagonal of A247495.

Cf. A001006, A247496.

Programs

Maple

S:= series(exp((n+1)*x)*BesselI(1,2*x)/x, x, 102):
seq(simplify(n!*coeff(S,x,n)),n=0..100); # Robert Israel, Nov 03 2017

Mathematica

Table[n! SeriesCoefficient[Exp[(n + 1) x] BesselI[1, 2 x]/x, {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[(1 - (n + 1) x - Sqrt[(1 - (n - 1) x) (1 - (n + 3) x)])/(2 x^2), {x, 0, n}], {n, 0, 19}]
Table[(n + 1)^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {2}, 4/(n + 1)^2], {n, 0, 19}]

Formula

a(n) = [x^n] (1 - (n + 1)*x - sqrt((1 - (n - 1)*x)*(1 - (n + 3)*x)))/(2*x^2).

a(n) ~ exp(1) * BesselI(1,2) * n^n. - Vaclav Kotesovec, Nov 13 2017