A220747 - OEIS

%I #22 Nov 03 2016 08:13:07

%S 1,3,15,105,315,3465,45045,135135,2297295,43648605,130945815,

%T 3011753745,75293843625,225881530875,6550564395375,203067496256625,

%U 609202488769875,21322087106945625,788917222956988125

%N a(n) = (2*n+1)!! / ((floor((n-1)/3)*2+1))!!

%C The a(n) appear in the analysis of a sequence that is related to the Madhava-Gregory-Leibniz formula for Pi, see A166107.

%F Limit_{n -> infinity} A166107(2*n)/a(2*n) = Pi.

%F Limit_{n -> infinity} A166107(2*n+1)/a(2*n+1) = -Pi.

%F E.g.f.: 2F2(5/6,7/6; 1/3,2/3; 4*x^3) + 3*x*(2F2(5/6,7/6; 2/3,4/3; 4*x^3) + 5*x*2F2(7/6,11/6; 4/3,5/3; 4*x^3)/2). - _Benedict W. J. Irwin_, Oct 19 2016

%p A220747 := n -> doublefactorial(2*n+1)/doublefactorial(A130823(n)): A130823 := n -> floor((n-1)/3)*2+1: seq(A220747(n), n=0..20);

%t Table[(2*n + 1)!!/((Floor[(n - 1)/3]*2 + 1))!!, {n, 0, 20}] (* _T. D. Noe_, Feb 26 2013 *)

%t CoefficientList[Series[HypergeometricPFQ[{5/6, 7/6}, {1/3, 2/3}, 4 x^3] + 3/2 x (2 HypergeometricPFQ[{5/6, 7/6}, {2/3, 4/3}, 4 x^3] + 5x HypergeometricPFQ[{7/6, 11/6}, {4/3, 5/3}, 4 x^3]), {x, 0, 20}], x]*Range[0, 20]! (* _Benedict W. J. Irwin_, Oct 19 2016 *)

%Y Cf. A000796, A001147, A130823, A166107.

%K nonn,easy

%O 0,2

%A _Johannes W. Meijer_, Feb 26 2013