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A293469
a(n) = Sum_{k=0..n} (2*k-1)!!*binomial(2*n-k, n).
2
1, 3, 12, 57, 330, 2436, 23226, 277389, 3966534, 65517210, 1220999208, 25279328958, 575024187192, 14247595540542, 381846383109030, 11004598454925405, 339324532631899110, 11146022446431209490, 388535338484934710040, 14324570939127320452350, 556887682690152668745660
OFFSET
0,2
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 09 2017
STATUS
approved