A271919 -id:A271919 - cp's OEIS frontend

A271926 Denominator of (Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6*j+1))) - 1).

1, 1, 13, 19, 95, 155, 5735, 49321, 345247, 11137, 97051, 175741, 12829093, 164988103, 164988103, 306406477, 2286263713, 235485162439, 25667882705851, 420784962391, 420784962391, 8773680484481, 166699929205139, 317414933691977, 16706049141683, 31931815448027, 5013295025340239

Offset: 1

N. J. A. Sloane, May 04 2016

			3, 5, 87/13, 156/19, 913/95, 1693/155, 69769/5735, 658529/49321, 5002953/345247, 173619/11137, 1616141/97051, 3107877/175741, 239756907/12829093, ...

J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math.CO/0211285, 2002.

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

f3:=proc(n) local j;
(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);

Table[Product[(2*j+1)*(3*j+4)/((j+1)*(6*j+1)),{j,0,n-1}]-1, {n,1,20}]//Denominator (* Vaclav Kotesovec, Oct 13 2017 *)

A271924 Denominator of (1/3)(Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6j+1))) - 1).

Original entry on oeis.org

1, 3, 13, 19, 285, 465, 17205, 147963, 345247, 11137, 291153, 175741, 12829093, 494964309, 494964309, 919219431, 6858791139, 706455487317, 77003648117553, 1262354887173, 1262354887173, 26321041453443, 500099787615417, 952244801075931, 50118147425049, 95795446344081

Offset: 1

N. J. A. Sloane, May 04 2016

			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, ...

J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math.CO/0211285, 2002.

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

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);

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

A271920 Denominator of Product_{j=1..n-1} ((3j+1)/(3j+2)).

Original entry on oeis.org

1, 5, 10, 11, 22, 187, 935, 1955, 391, 11339, 45356, 1334, 2668, 27347, 601634, 614713, 6147130, 162898945, 11847196, 12051458, 24102916, 30128645, 512186965, 7273054903, 7273054903, 80003603933, 400018019665, 809792576395, 9526971487, 77081860213, 1772882784899, 188604551585, 188604551585

Offset: 1

N. J. A. Sloane, May 04 2016

			1, 4/5, 7/10, 7/11, 13/22, 104/187, 494/935, 988/1955, 190/391, 5320/11339, 20615/45356, 589/1334, 1147/2668, 11470/27347, ...

J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math.CO/0211285, 2002-2003.

Cf. A271919 (numerators).

Other sequences of fractions from de Gier paper: A271921, A271922, A271923, A271924, A271925, A271926.

f:=proc(n) local j;
mul(((3*j+1)/(3*j+2)),j=1..n-1); end;
t1:=[seq(f(n),n=1..50)];
map(numer,t1);
map(denom,t1);

Table[Product[(3*j+1)/(3*j+2), {j, 1, n-1}] // Denominator, {n, 1, 33}] (* Jean-François Alcover, Mar 25 2018 *)

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

A271922 Denominator of nProduct_{j=1..n-1} ((3j + 1)/(3*j + 2)).

Original entry on oeis.org

1, 5, 10, 11, 22, 187, 935, 1955, 391, 11339, 45356, 667, 2668, 27347, 601634, 614713, 6147130, 162898945, 11847196, 6025729, 24102916, 30128645, 512186965, 7273054903, 7273054903, 80003603933, 400018019665, 809792576395, 9526971487, 77081860213, 1772882784899, 188604551585, 188604551585

Offset: 1

N. J. A. Sloane, May 04 2016

			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, ...

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

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

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);

Table[Denominator[n Product[(3j+1)/(3j+2), {j, 1, n-1}]], {n, 1, 33}] (* Jean-François Alcover, Dec 16 2018 *)

a(n) = denominator(n*prod(j=1, n-1, (3*j + 1)/(3*j + 2))); \\ Michel Marcus, Dec 16 2018

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

Original entry on oeis.org

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

N. J. A. Sloane, May 04 2016

			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, ...

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

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

Cf. A271924 (denominators), A073005, A186706.

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);

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 *)

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

A271925 Numerator of (Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6*j+1))) - 1).

Original entry on oeis.org

3, 5, 87, 156, 913, 1693, 69769, 658529, 5002953, 173619, 1616141, 3107877, 239756907, 3244922897, 3402714857, 6606018008, 51386679347, 5504537914811, 622652618545649, 10572475711004, 10931562934889, 235301799307039, 4608689892802861, 9034390134407023, 488936376609325, 959905250448181

Offset: 1

N. J. A. Sloane, May 04 2016

			3, 5, 87/13, 156/19, 913/95, 1693/155, 69769/5735, 658529/49321, 5002953/345247, 173619/11137, 1616141/97051, 3107877/175741, 239756907/12829093, ...

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

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

Cf. A271926 (denominators), A073005, A186706.

f3:=proc(n) local j;
(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);

Table[Product[(2*j+1)*(3*j+4)/((j+1)*(6*j+1)),{j,0,n-1}]-1, {n,1,20}]//Numerator (* Vaclav Kotesovec, Oct 13 2017 *)

a(n)/A271926(n) ~ c * (2*n)^(2/3), where c = Gamma(1/3)*3^(3/2)/(2*Pi) = 3*A073005/A186706. - Amiram Eldar, Aug 17 2025

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

Original entry on oeis.org

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

N. J. A. Sloane, May 04 2016

			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, ...

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

Cf. A271922 (denominators), A002161, A203145.

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

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);

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

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

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

cp's OEIS Frontend

A271926 Denominator of (Product_{j=0..n-1} (((2*j+1)*(3*j+4))/((j+1)*(6*j+1))) - 1).

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A271924 Denominator of (1/3)*(Product_{j=0..n-1} (((2*j+1)*(3*j+4))/((j+1)*(6*j+1))) - 1).

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A271920 Denominator of Product_{j=1..n-1} ((3*j+1)/(3*j+2)).

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A271922 Denominator of n*Product_{j=1..n-1} ((3*j + 1)/(3*j + 2)).

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

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A271925 Numerator of (Product_{j=0..n-1} (((2*j+1)*(3*j+4))/((j+1)*(6*j+1))) - 1).

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Maple

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

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Programs

Maple

Mathematica

PARI

Formula

A271926 Denominator of (Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6*j+1))) - 1).

A271924 Denominator of (1/3)(Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6j+1))) - 1).

A271920 Denominator of Product_{j=1..n-1} ((3j+1)/(3j+2)).

A271922 Denominator of nProduct_{j=1..n-1} ((3j + 1)/(3*j + 2)).

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

A271925 Numerator of (Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6*j+1))) - 1).

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