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A263529 Binomial transform of double factorial n!! (A006882). 3
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 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified February 23 00:18 EST 2019. Contains 320411 sequences. (Running on oeis4.)