A293468 a(n) = Sum_{k=0..n} k!*binomial(2*n-k, n).

1, 3, 11, 44, 189, 880, 4542, 26712, 182793, 1461368, 13477650, 140564536, 1627370146, 20621925504, 283161372284, 4182215376240, 66065933347425, 1111053154779720, 19814069772086730, 373435157945506680, 7415765258637418950, 154751460071567005920, 3385387828167428482020

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..449

Index entries for sequences related to factorial numbers

Formula

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

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

a(n) ~ exp(1) * n!. - Vaclav Kotesovec, Oct 18 2017

Maple

seq(simplify( GAMMA(n+1/2)*4^n*hypergeom([1, 1, -n], [-2*n], 1)/(sqrt(Pi)*n!)), n=0..30); # Robert Israel, Oct 09 2017

Mathematica

Table[Sum[k! Binomial[2 n - k, n], {k, 0, n}], {n, 0, 22}]
Table[Sum[Gamma[k + 1] Gamma[2 n - k  + 1]/(Gamma[n + 1] Gamma[n - k + 1]), {k, 0, n}], {n, 0, 22}]
Table[SeriesCoefficient[(1/(1 - x)^(n + 1)) 1/(1 + ContinuedFractionK[-Floor[(k + 1)/2] x, 1, {k, 1, n}]), {x, 0, n}], {n, 0, 22}]
Table[SeriesCoefficient[(1/(1 - x)^(n + 1)) Sum[k! x^k, {k, 0, n}], {x, 0, n}], {n, 0, 22}]
A293468[n_] := DifferenceRoot[Function[{a, k}, {(k+1)(k-n)a[k] + (k(n-2)-k^2+3n)
a[k+1] + (k-2n) a[k+2] == 0, a[0] == 0, a[1] == Binomial[2n, n]}]][1+n];
Table[A293468[n], {n, 0, 22}] (* Peter Luschny, Oct 09 2017 *)

Crossrefs

Cf. A000142, A000522, A003422, A014144, A092392, A270447, A293469.

Sequence in context: A302186 A151107 A063018 * A151108 A256752 A018963

Adjacent sequences: A293465 A293466 A293467 * A293469 A293470 A293471

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 09 2017

Status

approved