A103639 a(n) = Product_{i=1..2*n} (2*i+1).
1, 15, 945, 135135, 34459425, 13749310575, 7905853580625, 6190283353629375, 6332659870762850625, 8200794532637891559375, 13113070457687988603440625, 25373791335626257947657609375, 58435841445947272053455474390625, 157952079428395476360490147277859375
Offset: 0
Examples
Sequence starts 1, 1*3*5, 1*3*5*7*9, 1*3*5*7*9*11*13, ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..175
Programs
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Magma
[(n+1)*Factorial(2*n+1)*Catalan(2*n+1)/4^n: n in [0..20]]; // G. C. Greubel, Jan 29 2022
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Maple
A103639 := n -> pochhammer(1/2,2*n+1)*2^(2*n+1): seq(A103639(n), n=0..11); # Peter Luschny, Dec 19 2012
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Mathematica
Table[(4n+1)!!, {n, 0, 15}] (* Vladimir Reshetnikov, Nov 03 2015 *) -
PARI
vector(20, n, n--; prod(i=1, 2*n, 2*i+1)) \\ Altug Alkan, Nov 04 2015
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Sage
def A103639(n): return falling_factorial(4*n+2,2*n+1)*2^(-1-2*n) print([A103639(n) for n in (0..11)]) # Peter Luschny, Dec 14 2012
Formula
a(n) = (4*n+2)! / (2 * 4^n * (2*n+1)! ).
E.g.f.: sinh(x^2/2) = x^2/2! + 15*x^6/6! + 945*x^10/10! +...
a(n+1) = (4*n-1)*(4*n+1)*a(n), a(0) = 1.
a(n) = (4*n+1)!!. - Vladimir Reshetnikov, Nov 03 2015
a(n) = denominator((-3/2 - 2*n)!/sqrt(Pi)). - Peter Luschny, Jun 21 2020
D-finite with recurrence a(n) -(4*n-1)*(4*n+1)*a(n-1) = 0. - R. J. Mathar, Jun 06 2025
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