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
Keywords
Examples
G.f. = 1 + 2*x + 5*x^2 + 13*x^3 + 37*x^4 + 111*x^5 + 355*x^6 + 1191*x^7 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..795
- Eric Weisstein's MathWorld, Double Factorial.
Programs
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Mathematica
Table[Sum[k!!*Binomial[n, k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 20 2015 *) -
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
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