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A263801 Partial sums of odd double factorials (A001147) with alternating signs.

%I A263801 #17 Feb 16 2025 08:33:27
%S A263801 1,0,3,-12,93,-852,9543,-125592,1901433,-32557992,622171083,
%T A263801 -13127139492,303107003733,-7602746576892,205855300099983,
%U A263801 -5984428053529392,185914355908981233,-6146745514853869392,215496349961845902483,-7985298182676045656892
%N A263801 Partial sums of odd double factorials (A001147) with alternating signs.
%H A263801 G. C. Greubel, <a href="/A263801/b263801.txt">Table of n, a(n) for n = 0..400</a>
%H A263801 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>.
%H A263801 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>.
%H A263801 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/Erf.html">Erf</a>.
%F A263801 a(n) = Sum_{k=0..n} (-1)^k*(2*k-1)!!.
%F A263801 a(n) = (sqrt(Pi)*erfc(1/sqrt(2))-Gamma(-n-1/2, 1/2)*(2*n+1)!!/(-2)^(n+1))*exp(1/2)/sqrt(2), where Gamma(a, x) is the upper incomplete Gamma function.
%F A263801 E.g.f.: 1/sqrt(2*x+1)+sqrt(Pi/2)*exp(x+1/2)*(erf(sqrt(x+1/2))-erf(1/sqrt(2))).
%F A263801 Recurrence: a(0) = 1, a(1) = 0, a(n+2) = (2*n+3)*a(n)-(2*n+2)*a(n+1).
%F A263801 0 = a(n)*(-2*a(n+1) + a(n+2) + a(n+3)) + a(n+1)*(+3*a(n+1) - 3*a(n+2) - a(n+3)) + a(n+2)*(+a(n+2)) if n>=0. - _Michael Somos_, Oct 30 2015
%e A263801 For n = 4, a(4) = Sum_{k=0..4} (-1)^k*(2*k-1)!! = (-1)!! - 1!! + 3!! - 5!! + 7!! = 1 - 1 + 3 - 15 + 105 = 93.
%e A263801 G.f. = 1 + 3*x^2 - 12*x^3 + 93*x^4 - 852*x^5 + 9543*x^6 - 125592*x^7 + ...
%t A263801 Table[Sum[(-1)^k (2k-1)!!, {k, 0, n}], {n, 0, 20}]
%t A263801 Round@Table[(Sqrt[Pi] Erfc[1/Sqrt[2]] - Gamma[-n-1/2, 1/2] (2n+1)!!/(-2)^(n+1)) Sqrt[E/2], {n, 0, 20}]
%o A263801 (PARI) for(n=0,50, print1(sum(k=0,n, (-1)^k*(2*k)!/(2^k*k!)), ", ")) \\ _G. C. Greubel_, Apr 08 2017
%Y A263801 Cf. A001147, A076795.
%K A263801 sign
%O A263801 0,3
%A A263801 _Vladimir Reshetnikov_, Oct 26 2015