A293469 a(n) = Sum_{k=0..n} (2*k-1)!!*binomial(2*n-k, n).

1, 3, 12, 57, 330, 2436, 23226, 277389, 3966534, 65517210, 1220999208, 25279328958, 575024187192, 14247595540542, 381846383109030, 11004598454925405, 339324532631899110, 11146022446431209490, 388535338484934710040, 14324570939127320452350, 556887682690152668745660

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..403

Index entries for sequences related to factorial numbers

Formula

a(n) = [x^n] 1/((1 - x)^(n+1)*(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - 6*x/(1 - ...)))))))), a continued fraction.

a(n) = Gamma(n+1/2)*hypergeom([1/2, 1, -n], [-2*n], 2)*4^n/(n!*sqrt(Pi)). - Robert Israel, Oct 09 2017

a(n) ~ 2^(n + 1/2) * n^n / exp(n - 1/2). - Vaclav Kotesovec, Oct 18 2017

Maple

seq(add(doublefactorial(2*k-1)*binomial(2*n-k, n), k=0..n), n=0..40); # Robert Israel, Oct 09 2017

Mathematica

Table[Sum[(2 k - 1)!! Binomial[2 n - k, n], {k, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[(1/(1 - x)^(n + 1)) 1/(1 + ContinuedFractionK[-k x, 1, {k, 1, n}]), {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[(1/(1 - x)^(n + 1)) Sum[(2 k - 1)!! x^k, {k, 0, n}], {x, 0, n}], {n, 0, 20}]

Crossrefs

Cf. A001147, A076795, A084262, A092392, A270447, A293468.

Sequence in context: A307495 A302101 A279271 * A009248 A012709 A370528

Adjacent sequences: A293466 A293467 A293468 * A293470 A293471 A293472

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 09 2017

Status

approved