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

%I A263529 #28 Feb 16 2025 08:33:27
%S A263529 1,2,5,13,37,111,355,1191,4201,15445,59171,234983,966397,4101709,
%T A263529 17946783,80754331,373286481,1769440513,8592681907,42689422871,
%U A263529 216789872741,1124107246669,5947013363479,32071798826115,176194545585529,985330955637801,5605802379087067
%N A263529 Binomial transform of double factorial n!! (A006882).
%H A263529 G. C. Greubel, <a href="/A263529/b263529.txt">Table of n, a(n) for n = 0..795</a>
%H A263529 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>.
%F A263529 a(n) = Sum_{k=0..n} k!!*binomial(n,k), where k!! = A006882(k).
%F A263529 Sum_{k=0..n} (-1)^(k+n)*a(k)*binomial(n,k) = n!!.
%F A263529 E.g.f.: exp(x) + exp((2*x+x^2)/2)*(2 + sqrt(2*Pi)*erf(x/sqrt(2)))*x/2.
%F A263529 Recurrence: (n+1)*a(n+2) = (n+2)*a(n+1) + (n+1)*(n+2)*a(n) - 1.
%F A263529 a(n) ~ (sqrt(2) + sqrt(Pi))/2 * n^(n/2 + 1/2) * exp(sqrt(n) - n/2 - 1/4). - _Vaclav Kotesovec_, Oct 20 2015
%F A263529 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
%F A263529 G.f.: Sum_{k>=0} k!!*x^k/(1 - x)^(k+1). - _Ilya Gutkovskiy_, Apr 12 2019
%e A263529 G.f. = 1 + 2*x + 5*x^2 + 13*x^3 + 37*x^4 + 111*x^5 + 355*x^6 + 1191*x^7 + ...
%t A263529 Table[Sum[k!!*Binomial[n, k], {k, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, Oct 20 2015 *)
%o A263529 (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
%Y A263529 Cf. A006882, A262020.
%K A263529 nonn
%O A263529 0,2
%A A263529 _Vladimir Reshetnikov_, Oct 19 2015