A271923 Numerator of (1/3)*(Product_{j=0..n-1} (((2*j+1)*(3*j+4))/((j+1)*(6*j+1))) - 1).

1, 5, 29, 52, 913, 1693, 69769, 658529, 1667651, 57873, 1616141, 1035959, 79918969, 3244922897, 3402714857, 6606018008, 51386679347, 5504537914811, 622652618545649, 10572475711004, 10931562934889, 235301799307039, 4608689892802861, 9034390134407023, 488936376609325, 959905250448181

Offset: 1

Author

N. J. A. Sloane, May 04 2016

Keywords

• nonn
• frac

Examples

			1, 5/3, 29/13, 52/19, 913/285, 1693/465, 69769/17205, 658529/147963, 1667651/ 345247, 57873/11137, 1616141/291153, 1035959/175741, 79918969/12829093, ...
		

Links

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

Crossrefs

Sequences of fractions from de Gier paper: A271919-A271926.

Cf. A271924 (denominators), A073005, A186706.

Programs

Maple

f3:=proc(n) local j;
(1/3)*(mul(((2*j+1)*(3*j+4))/((j+1)*(6*j+1)),j=0..n-1)-1); end;
t3:=[seq(f3(n),n=1..50)];
map(numer,t3);
map(denom,t3);

Mathematica

a[n_] := (1/3)*(Product[((2*j + 1)*(3*j + 4))/((j + 1)*(6*j + 1)), {j, 0, n - 1}] - 1) // Numerator;
Array[a, 26] (* Jean-François Alcover, Nov 30 2017 *)

Formula

a(n)/A271924(n) ~ c * (2*n)^(2/3), where c = Gamma(1/3)*sqrt(3)/(2*Pi) = A073005/A186706. - Amiram Eldar, Aug 17 2025