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

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..175

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

Example

Sequence starts 1, 1*3*5, 1*3*5*7*9, 1*3*5*7*9*11*13, ...

Maple

A103639 := n -> pochhammer(1/2, 2*n+1)*2^(2*n+1):
seq(A103639(n), n=0..11); # Peter Luschny, Dec 19 2012

Mathematica

Table[(4n+1)!!, {n, 0, 15}] (* Vladimir Reshetnikov, Nov 03 2015 *)

Prog

(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
(PARI) vector(20, n, n--; prod(i=1, 2*n, 2*i+1)) \\ Altug Alkan, Nov 04 2015
(Magma) [(n+1)*Factorial(2*n+1)*Catalan(2*n+1)/4^n: n in [0..20]]; // G. C. Greubel, Jan 29 2022

Crossrefs

Bisection of the double factorials A001147.

Cf. A102992.

Cf. Odd part of A024343 and A009564.

Sequence in context: A261067 A136419 A231121 * A055413 A067408 A274713

Adjacent sequences: A103636 A103637 A103638 * A103640 A103641 A103642

Keyword

nonn

Author

Ralf Stephan, Feb 18 2005

Status

approved