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A103639 a(n) = Product_{i=1..2*n} (2*i+1).

%I A103639 #41 Aug 31 2025 02:58:04
%S A103639 1,15,945,135135,34459425,13749310575,7905853580625,6190283353629375,
%T A103639 6332659870762850625,8200794532637891559375,
%U A103639 13113070457687988603440625,25373791335626257947657609375,58435841445947272053455474390625,157952079428395476360490147277859375
%N A103639 a(n) = Product_{i=1..2*n} (2*i+1).
%H A103639 G. C. Greubel, <a href="/A103639/b103639.txt">Table of n, a(n) for n = 0..175</a>
%F A103639 a(n) = (4*n+2)! / (2 * 4^n * (2*n+1)! ).
%F A103639 E.g.f.: sinh(x^2/2) = x^2/2! + 15*x^6/6! + 945*x^10/10! +...
%F A103639 a(n+1) = (4*n-1)*(4*n+1)*a(n), a(0) = 1.
%F A103639 a(n) = (4*n+1)!!. - _Vladimir Reshetnikov_, Nov 03 2015
%F A103639 a(n) = denominator((-3/2 - 2*n)!/sqrt(Pi)). - _Peter Luschny_, Jun 21 2020
%F A103639 D-finite with recurrence a(n) -(4*n-1)*(4*n+1)*a(n-1) = 0. - _R. J. Mathar_, Jun 06 2025
%F A103639 Sum_{n>=0} 1/a(n) = (sqrt(Pi/2)/2) * (e * erf(1/sqrt(2)) + erfi(1/sqrt(2)) / e). - _Amiram Eldar_, Aug 31 2025
%e A103639 Sequence starts 1, 1*3*5, 1*3*5*7*9, 1*3*5*7*9*11*13, ...
%p A103639 A103639 := n -> pochhammer(1/2,2*n+1)*2^(2*n+1):
%p A103639 seq(A103639(n), n=0..11); # _Peter Luschny_, Dec 19 2012
%t A103639 Table[(4n+1)!!, {n, 0, 15}] (* _Vladimir Reshetnikov_, Nov 03 2015 *)
%o A103639 (Sage)
%o A103639 def A103639(n):
%o A103639     return falling_factorial(4*n+2,2*n+1)*2^(-1-2*n)
%o A103639 print([A103639(n) for n in (0..11)])  # _Peter Luschny_, Dec 14 2012
%o A103639 (PARI) vector(20, n, n--; prod(i=1, 2*n, 2*i+1)) \\ _Altug Alkan_, Nov 04 2015
%o A103639 (Magma) [(n+1)*Factorial(2*n+1)*Catalan(2*n+1)/4^n: n in [0..20]]; // _G. C. Greubel_, Jan 29 2022
%Y A103639 Bisection of the double factorials A001147.
%Y A103639 Cf. A102992.
%Y A103639 Odd part of A024343 and A009564.
%K A103639 nonn,changed
%O A103639 0,2
%A A103639 _Ralf Stephan_, Feb 18 2005