A271921 Numerator of n*Product_{j=1..n-1} ((3*j + 1)/(3*j + 2)).

1, 8, 21, 28, 65, 624, 3458, 7904, 1710, 53200, 226765, 3534, 14911, 160580, 3699075, 3945680, 41084393, 1131029172, 85276009, 44882110, 185464461, 239133664, 4187556548, 61174739136, 62862555700, 709808057504, 3639472564077, 7548535688456, 90908444753, 752345749680, 17686394665394

Offset: 1

Author

N. J. A. Sloane, May 04 2016

Keywords

• nonn
• frac

Examples

			1, 8/5, 21/10, 28/11, 65/22, 624/187, 3458/935, 7904/1955, 1710/391, 53200/ 11339, 226765/45356, 3534/667, 14911/2668, 160580/27347, 3699075/601634, ...
		

Links

• Jan de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.

Crossrefs

Cf. A271922 (denominators), A002161, A203145.

Sequences of fractions from de Gier paper: A271919, A271920, A271922, A271923, A271924, A271925, A271926.

Programs

Maple

f:=proc(n) local j;
mul(((3*j+1)/(3*j+2)),j=1..n-1); end;
t2:=[seq(n*f(n),n=1..50)];
map(numer,t2);
map(denom,t2);

Mathematica

Table[n*Product[(3*j+1)/(3*j+2), {j, 1, n-1}] // Numerator, {n, 1, 31}] (* Jean-François Alcover, Mar 25 2018 *)

PARI

a(n) = numerator(n*prod(j=1, n-1, (3*j + 1)/(3*j + 2))); \\ Michel Marcus, Mar 25 2018

Formula

a(n)/A271922(n) ~ c * (2*n)^(2/3), where c = Gamma(5/6)/sqrt(Pi) = A203145/A002161. - Amiram Eldar, Aug 17 2025