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

%I

%S 1,2,10,76,777,9996,155139,2821400,58856963,1385621260,36343079188,

%T 1051024082472,33226817252215,1140040324751160,42193259673938754,

%U 1675570154136359472,71069261432474378715,3206616936773061141900,153358034674756782660342,7749560706936442485607560

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

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

%H Robert Israel, <a href="/A294573/b294573.txt">Table of n, a(n) for n = 0..300</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

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

%F a(n) ~ exp(1) * BesselI(1,2) * n^n. - _Vaclav Kotesovec_, Nov 13 2017

%p S:= series(exp((n+1)*x)*BesselI(1,2*x)/x, x, 102):

%p seq(simplify(n!*coeff(S,x,n)),n=0..100); # _Robert Israel_, Nov 03 2017

%t Table[n! SeriesCoefficient[Exp[(n + 1) x] BesselI[1, 2 x]/x, {x, 0, n}], {n, 0, 19}]

%t Table[SeriesCoefficient[(1 - (n + 1) x - Sqrt[(1 - (n - 1) x) (1 - (n + 3) x)])/(2 x^2), {x, 0, n}], {n, 0, 19}]

%t Table[(n + 1)^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {2}, 4/(n + 1)^2], {n, 0, 19}]

%Y Diagonal of A247495.

%Y Cf. A001006, A247496.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Nov 02 2017

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Last modified November 14 02:19 EST 2019. Contains 329108 sequences. (Running on oeis4.)