A263801 Partial sums of odd double factorials (A001147) with alternating signs.
1, 0, 3, -12, 93, -852, 9543, -125592, 1901433, -32557992, 622171083, -13127139492, 303107003733, -7602746576892, 205855300099983, -5984428053529392, 185914355908981233, -6146745514853869392, 215496349961845902483, -7985298182676045656892
Offset: 0
Keywords
Examples
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.
G.f. = 1 + 3*x^2 - 12*x^3 + 93*x^4 - 852*x^5 + 9543*x^6 - 125592*x^7 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..400
- Eric Weisstein's MathWorld, Double Factorial.
- Eric Weisstein's MathWorld, Incomplete Gamma Function.
- Eric Weisstein's MathWorld, Erf.
Programs
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Mathematica
Table[Sum[(-1)^k (2k-1)!!, {k, 0, n}], {n, 0, 20}] 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}] -
PARI
for(n=0,50, print1(sum(k=0,n, (-1)^k*(2*k)!/(2^k*k!)), ", ")) \\ G. C. Greubel, Apr 08 2017
Formula
a(n) = Sum_{k=0..n} (-1)^k*(2*k-1)!!.
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.
E.g.f.: 1/sqrt(2*x+1)+sqrt(Pi/2)*exp(x+1/2)*(erf(sqrt(x+1/2))-erf(1/sqrt(2))).
Recurrence: a(0) = 1, a(1) = 0, a(n+2) = (2*n+3)*a(n)-(2*n+2)*a(n+1).
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