A263529 Binomial transform of double factorial n!! (A006882).

1, 2, 5, 13, 37, 111, 355, 1191, 4201, 15445, 59171, 234983, 966397, 4101709, 17946783, 80754331, 373286481, 1769440513, 8592681907, 42689422871, 216789872741, 1124107246669, 5947013363479, 32071798826115, 176194545585529, 985330955637801, 5605802379087067

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..795

Eric Weisstein's MathWorld, Double Factorial.

Formula

a(n) = Sum_{k=0..n} k!!*binomial(n,k), where k!! = A006882(k).

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

E.g.f.: exp(x) + exp((2*x+x^2)/2)*(2 + sqrt(2*Pi)*erf(x/sqrt(2)))*x/2.

Recurrence: (n+1)*a(n+2) = (n+2)*a(n+1) + (n+1)*(n+2)*a(n) - 1.

a(n) ~ (sqrt(2) + sqrt(Pi))/2 * n^(n/2 + 1/2) * exp(sqrt(n) - n/2 - 1/4). - Vaclav Kotesovec, Oct 20 2015

0 = a(n)*(+a(n+1) - 2*a(n+2) - 2*a(n+3) + a(n+4)) + a(n+1)*(+3*a(n+2) + a(n+3) - a(n+4)) + a(n+2)*(-2*a(n+2) + a(n+3)) for all n>=0. - Michael Somos, Oct 20 2015

G.f.: Sum_{k>=0} k!!*x^k/(1 - x)^(k+1). - Ilya Gutkovskiy, Apr 12 2019

Example

G.f. = 1 + 2*x + 5*x^2 + 13*x^3 + 37*x^4 + 111*x^5 + 355*x^6 + 1191*x^7 + ...

Mathematica

Table[Sum[k!!*Binomial[n, k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 20 2015 *)

Prog

(PARI) vector(50, n, n--; sum(k=0, n, prod(i=0, (k-1)\2, k - 2*i)*binomial(n, k))) \\ Altug Alkan, Oct 20 2015

Crossrefs

Cf. A006882, A262020.

Sequence in context: A003080 A149854 A151442 * A053732 A216617 A243412

Adjacent sequences: A263526 A263527 A263528 * A263530 A263531 A263532

Keyword

nonn

Author

Vladimir Reshetnikov, Oct 19 2015

Status

approved