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A076795 Partial sums of (2n-1)!!. 5

%I #31 Oct 29 2015 22:49:55

%S 0,1,2,5,20,125,1070,11465,146600,2173625,36633050,691362125,

%T 14440672700,330674815925,8236528396550,221694575073425,

%U 6411977928702800,198310761891213425,6530970632654064050

%N Partial sums of (2n-1)!!.

%H Vincenzo Librandi, <a href="/A076795/b076795.txt">Table of n, a(n) for n = 0..300</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html">Incomplete Gamma Function</a>.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Erfi.html">Erfi</a>.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/DawsonsIntegral.html">Dawson's Integral</a>.

%F E.g.f.: exp(x) * Integral_{t=0, x} exp(-t) / sqrt(1-2*t) dt.

%F a(n) = a(n-1) * (2*n - 2) - a(n-2) * (2*n - 3).

%F a(n) ~ 1/(sqrt(2) * n) * 2^n * (n/e)^n.

%F G.f.: A(x)=x/(1-x)*(1 + x/(U(0)-x)), where U(k)= (2*k+1)*x + 1 - (2*k+3)*x/U(k+1); (continued fraction Euler's 1st kind, 1-step ). - Sergei N. Gladkovskii, Jun 27 2012

%F G.f.: x/(1-x)/Q(0), where Q(k)= 1 - x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 19 2013

%F G.f.: G(0)*x/(1-x), where G(k)= 1 - x*(k+1)/(x*(k+1) - 1/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Aug 04 2013

%F From _Vladimir Reshetnikov_, Oct 27 2015: (Start)

%F a(n) = Sum_{k=0..n-1} (2*k-1)!!.

%F a(n) = sqrt(Pi/2)*exp(-1/2)*erfi(1/sqrt(2))+(2*n-1)!!*Im(Gamma(1/2-n, -1/2))*exp(-1/2)/((-2)^n*sqrt(2)), where Gamma(a, x) is the upper incomplete Gamma function.

%F E.g.f.: sqrt(2)*(exp(x)*F(1/sqrt(2))-F(sqrt(1/2-x))), where F(x)=exp(-x^2)*erfi(x)*sqrt(Pi)/2 is the Dawson integral. (End)

%F 0 = a(n)*(+2*a(n+1) - 3*a(n+2) + a(n+3)) + a(n+1)*(-a(n+1) + a(n+2) - a(n+3)) + a(n+2)*(+a(n+2)) for all n>=0. - _Michael Somos_, Oct 29 2015

%e G.f. = x + 2*x^2 + 5*x^3 + 20*x^4 + 125*x^5 + 1070*x^6 + 11465*x^7 + 146600*x^8 + ...

%t Join[{0},Accumulate[Table[(2n-1)!!,{n,0,20}]]] (* _Harvey P. Dale_, Jan 27 2013 *)

%t Round@Table[Sqrt[Pi/(2E)] Erfi[1/Sqrt[2]] + (2n-1)!! Im[Gamma[1/2-n, -1/2]]/((-2)^n Sqrt[2E]), {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 27 2015 *)

%t a[ n_] := Sum[ (2 k - 1)!!, {k, 0, n - 1}]; (* _Michael Somos_, Oct 29 2015 *)

%o (PARI) {a(n) = if( n<0, 0, sum(k=0, n-1, (2*k)! / (k! * 2^k)))};

%Y Cf. A001147, A263801.

%K nonn

%O 0,3

%A _Michael Somos_, Nov 16 2002

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)