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A294642 a(n) = n! * [x^n] exp(n*x)*BesselI(1,2*sqrt(2)*x)/(sqrt(2)*x). 0
1, 1, 6, 45, 456, 5825, 89896, 1627437, 33822944, 793783233, 20765009344, 599157626925, 18904594000128, 647524807918209, 23929038677825152, 948995910652193325, 40203601321988822528, 1812025020244371552897, 86577002960871477916672, 4371100278517527047687213 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..19.

FORMULA

a(n) = [x^n] (1 - n*x - sqrt(1 - 2*n*x + (n^2 - 8)*x^2))/(4*x^2).

a(n) = [x^n] 1/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - n*x - 2*x^2/(1 - ...))))), a continued fraction.

a(n) = Sum_{k=0..floor(n/2)} 2^k*n^(n-2*k)*binomial(n,2*k)*A000108(k).

a(n) = n^n*2F1(1/2-n/2,-n/2; 2; 8/n^2).

a(n) ~ c * n^n, where c = BesselI(1, 2*sqrt(2))/sqrt(2) = 2.3948330992734... - Vaclav Kotesovec, Nov 06 2017

MATHEMATICA

Simplify[Table[n! SeriesCoefficient[Exp[n x] BesselI[1, 2 Sqrt[2] x]/(Sqrt[2] x), {x, 0, n}], {n, 0, 19}]]

Table[SeriesCoefficient[(1 - n x - Sqrt[1 - 2 n x + (n^2 - 8) x^2])/(4 x^2), {x, 0, n}], {n, 0, 19}]

Table[SeriesCoefficient[1/(1 - n x + ContinuedFractionK[-2 x^2, 1 - n x, {i, 1, n}]), {x, 0, n}], {n, 0, 19}]

Join[{1}, Table[Sum[2^k n^(n - 2 k) Binomial[n, 2 k] CatalanNumber[k], {k, 0, Floor[n/2]}], {n, 1, 19}]]

Join[{1}, Table[n^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {2}, 8/n^2], {n, 1, 19}]]

CROSSREFS

Cf. A000108, A001003, A025235, A068764, A071356, A151374, A247496, A292716.

Sequence in context: A228194 A084064 A186925 * A109516 A245493 A078865

Adjacent sequences:  A294639 A294640 A294641 * A294643 A294644 A294645

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Nov 05 2017

STATUS

approved

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Last modified December 12 20:04 EST 2017. Contains 295954 sequences.