A214776 -id:A214776 - cp's OEIS frontend

A174687 Central coefficients T(2n,n) of the Catalan triangle A033184.

1, 2, 9, 48, 275, 1638, 9996, 62016, 389367, 2466750, 15737865, 100975680, 650872404, 4211628008, 27341497800, 177996090624, 1161588834303, 7596549816030, 49772989810635, 326658445806000, 2147042307851595, 14130873926790390, 93115841412899760

Offset: 0

Paul Barry, Mar 27 2010

A033184 is the Riordan array (c(x), x*c(x)), c(x) the g.f. of A000108.
Number of standard Young tableaux of shape [2n, n]. Also the number of binary words with 2n 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's. The a(2) = 9 words are: 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100. - Alois P. Heinz, Aug 15 2012
Number of lattice paths from (0,0) to (2n,n) not above y=x. - Ran Pan, Apr 08 2015

Alois P. Heinz, Table of n, a(n) for n = 0..370
Paul Barry, On the Central Coefficients of Riordan Matrices, Journal of Integer Sequences, 16 (2013), Article 13.5.1.
Dmitry Kruchinin and Vladimir Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), Article 12.9.3.
Ran Pan, Exercise L, Project P
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.

Cf. A000108, A001764, A033184.

Column k=2 of A214776.

[(n+1)*Binomial(3*n,n)/(2*n+1): n in [0..25]]; // Vincenzo Librandi, Apr 08 2015

a:= n-> binomial(3*n,n)*(n+1)/(2*n+1):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 15 2012

Table[Binomial[3n,n](n+1)/(2n+1), {n, 0, 25}] (* Vincenzo Librandi, Apr 08 2015 *)

a(n) = (n+1)*binomial(3*n,n)/(2*n+1); \\ Michel Marcus, Nov 12 2022

[(n+1)*binomial(3*n,n)/(2*n+1) for n in range(31)] # G. C. Greubel, Nov 09 2022

a(n) = (n+1)*C(3*n, n)/(2n+1) = (n+1)*[x^(n+1)]( Rev(x/c(x)) ) = (n+1)*A001764(n), c(x) the g.f. of A000108.
G.f.: A(x) = sin(arcsin((3^(3/2)*sqrt(x))/2)/3)/(sqrt(3)*sqrt(x)) + cos(arcsin((3^(3/2)* sqrt(x))/2)/3)/(2*sqrt(1-(27*x)/4)). - Vladimir Kruchinin, May 25 2012
2*n*(2*n+1)*a(n) = 3*(13*n^2 -10*n +1)*a(n-1) -9*(3*n-4)*(3*n-5)*a(n-2). - R. J. Mathar, Nov 24 2012
a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(n+1). - Ilya Gutkovskiy, Nov 01 2017
a(n) ~ 3^(3*n+1/2) / (4^(n+1) * sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025

A126596 a(n) = binomial(4n,n)(2n+1)/(3n+1).

Original entry on oeis.org

1, 3, 20, 154, 1260, 10659, 92092, 807300, 7152444, 63882940, 574221648, 5188082354, 47073334100, 428634152730, 3914819231400, 35848190542920, 329007937216860, 3025582795190340, 27872496751392496, 257172019222240200, 2376196095585231920, 21983235825545286435

Offset: 0

Philippe Deléham, Mar 13 2007

Number of standard Young tableaux of shape [3n,n]. Also the number of binary words with 3n 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's. The a(1) = 3 words are: 1011, 1101, 1110. - Alois P. Heinz, Aug 15 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Column k=3 of A214776.

a126596 n = a005810 n * a005408 n `div` a016777 n
-- Reinhard Zumkeller, Mar 04 2012

[Binomial(4*n,n)*(2*n+1)/(3*n+1): n in [0..20]]; // Vincenzo Librandi, Nov 18 2011

seq((2*n+1)*binomial(4*n,n)/(3*n+1),n=0..22); # Emeric Deutsch, Mar 27 2007

Table[(Binomial[4n,n](2n+1))/(3n+1),{n,0,30}] (* Harvey P. Dale, Feb 06 2016 *)

a(n) = A039599(2*n,n).
a(n) = (2*n+1)*A002293(n). - Mark van Hoeij, Nov 17 2011
a(n) = A208983(2*n+1). - Reinhard Zumkeller, Mar 04 2012
a(n) = A005810(n) * A005408(n) / A016777(n). - Reinhard Zumkeller, Mar 04 2012
a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(2*n+1). - Ilya Gutkovskiy, Nov 01 2017
Recurrence: 3*n*(3*n-1)*(3*n+1)*a(n) = 8*(2*n+1)*(4*n-3)*(4*n-1)*a(n-1). - Vaclav Kotesovec, Feb 03 2018
a(n) ~ 2^(8*n+3/2) / (3^(3*n+3/2) * sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025

More terms from Emeric Deutsch, Mar 27 2007

A215541 a(n) = binomial(5n,n)(3n+1)/(4n+1).

Original entry on oeis.org

1, 4, 35, 350, 3705, 40480, 451269, 5101360, 58261125, 670609940, 7766844470, 90404916420, 1056658719675, 12393263030400, 145787921878840, 1719353829326880, 20322351313767965, 240674861588534100, 2855214354095519625, 33924757188414045330, 403641797464597415570

Offset: 0

Alois P. Heinz, Aug 15 2012

Number of standard Young tableaux of shape [4n,n]. Also the number of binary words with 4n 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's. The a(1) = 4 words are: 10111, 11011, 11101, 11110.

Alois P. Heinz, Table of n, a(n) for n = 0..285
Wikipedia, Young tableau.

Column k=4 of A214776.

a:= n-> binomial(5*n,n)*(3*n+1)/(4*n+1):
seq(a(n), n=0..25);

a[n_] := Binomial[5*n,n]*(3*n+1)/(4*n+1); Array[a, 21, 0] (* Amiram Eldar, Aug 29 2025 *)

a(n) = C(5*n,n)*(3*n+1)/(4*n+1).
a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(3*n+1). - Ilya Gutkovskiy, Nov 01 2017
Recurrence: 8*n*(2*n - 1)*(3*n - 2)*(4*n - 1)*(4*n + 1)*a(n) = 5*(3*n + 1)*(5*n - 4)*(5*n - 3)*(5*n - 2)*(5*n - 1)*a(n-1). - Vaclav Kotesovec, Feb 03 2018
a(n) ~ 3 * 5^(5*n+1/2) / (2^(8*n+7/2) * sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025

A215542 a(n) = binomial(6n,n)(4n+1)/(5n+1).

Original entry on oeis.org

1, 5, 54, 663, 8602, 115101, 1570800, 21732542, 303719922, 4277470470, 60610884906, 863102246760, 12340998865104, 177064708142315, 2547927647834040, 36757054103054076, 531436857842656610, 7698470087956704210, 111712846834848074340, 1623556455926349703605

Offset: 0

Alois P. Heinz, Aug 15 2012

Number of standard Young tableaux of shape [5n,n].

Alois P. Heinz, Table of n, a(n) for n = 0..260
Wikipedia, Young tableau.

Column k=5 of A214776.

a:= n-> binomial(6*n,n)*(4*n+1)/(5*n+1):
seq(a(n), n=0..20);

Table[Binomial[6n,n] (4n+1)/(5n+1),{n,0,30}] (* Harvey P. Dale, Mar 06 2014 *)

a(n) = C(6*n,n)*(4*n+1)/(5*n+1).
a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(4*n+1). - Ilya Gutkovskiy, Nov 01 2017
Recurrence: 5*n*(4*n - 3)*(5*n - 3)*(5*n - 2)*(5*n - 1)*(5*n + 1)*a(n) = 72*(2*n - 1)*(3*n - 2)*(3*n - 1)*(4*n + 1)*(6*n - 5)*(6*n - 1)*a(n-1). - Vaclav Kotesovec, Feb 03 2018
a(n) ~ 3^(6*n+1/2) * 4^(3*n+1) / (5^(5*n+3/2) * sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025

A215543 Number of standard Young tableaux of shape [3n,3].

Original entry on oeis.org

0, 5, 48, 154, 350, 663, 1120, 1748, 2574, 3625, 4928, 6510, 8398, 10619, 13200, 16168, 19550, 23373, 27664, 32450, 37758, 43615, 50048, 57084, 64750, 73073, 82080, 91798, 102254, 113475, 125488, 138320, 151998, 166549, 182000, 198378, 215710, 234023, 253344

Offset: 0

Alois P. Heinz, Aug 15 2012

Also the number of binary words with 3n 1's and 3 0's such that for every prefix the number of 1's is >= the number of 0's. The a(1) = 5 words are: 101010, 101100, 110010, 110100, 111000.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Young tableau.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Row n=3 of A214776.

a:= n-> max(0, (3*n-2)*(3*n+2)*(n+1)/2):
seq(a(n), n=0..40);

LinearRecurrence[{4,-6,4,-1},{0,5,48,154,350},50] (* Harvey P. Dale, Dec 16 2017 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,4,-6,4]^n*[0;5;48;154])[1,1] \\ Charles R Greathouse IV, Oct 18 2022

G.f.: (2*x^3-8*x^2+28*x+5)*x/(x-1)^4.
a(n) = (3*n-2)*(3*n+2)*(n+1)/2 for n>0, a(0) = 0.
From Amiram Eldar, Aug 29 2025: (Start)
Sum_{n>=1} 1/a(n) = 7/20 + sqrt(3)*Pi/10 - 3*log(3)/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/5 - 4*log(2)/5 - 7/20. (End)

A215544 Number of standard Young tableaux of shape [4n,4].

Original entry on oeis.org

0, 14, 275, 1260, 3705, 8602, 17199, 31000, 51765, 81510, 122507, 177284, 248625, 339570, 453415, 593712, 764269, 969150, 1212675, 1499420, 1834217, 2222154, 2668575, 3179080, 3759525, 4416022, 5154939, 5982900, 6906785, 7933730, 9071127, 10326624, 11708125

Offset: 0

Alois P. Heinz, Aug 15 2012

Also the number of binary words with 4n 1's and 4 0's such that for every prefix the number of 1's is >= the number of 0's. The a(1) = 14 words are: 10101010, 10101100, 10110010, 10110100, 10111000, 11001010, 11001100, 11010010, 11010100, 11011000, 11100010, 11100100, 11101000, 11110000.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Young tableau.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Row n=4 of A214776.

a:= n-> max(0, (4*n+3)*(2*n+1)*(4*n-3)*(n+1)/3):
seq(a(n), n=0..40);

LinearRecurrence[{5,-10,10,-5,1},{0,14,275,1260,3705,8602},40] (* Harvey P. Dale, Jan 25 2024 *)

a(n)=max((4*n-3)*(4*n+3)*(2*n+1)*(n+1)/3,0) \\ Charles R Greathouse IV, Oct 21 2022

G.f.: (3*x^4-15*x^3-25*x^2-205*x-14)*x/(x-1)^5.
a(n) = (4*n-3)*(4*n+3)*(2*n+1)*(n+1)/3 for n>0, a(0) = 0.
Sum_{n>=1} 1/a(n) = 31/105 - 17*Pi/35 + 66*log(2)/35. - Amiram Eldar, Aug 29 2025

A215545 Number of standard Young tableaux of shape [5n,5].

Original entry on oeis.org

0, 42, 1638, 10659, 40480, 115101, 272272, 566618, 1072764, 1888460, 3137706, 4973877, 7582848, 11186119, 16043940, 22458436, 30776732, 41394078, 54756974, 71366295, 91780416, 116618337, 146562808, 182363454, 224839900, 274884896, 333467442, 401635913

Offset: 0

Alois P. Heinz, Aug 15 2012

Also the number of binary words with 5n 1's and 5 0's such that for every prefix the number of 1's is >= the number of 0's.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Young tableau
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Row n=5 of A214776.

a:= n-> max(0, (5*n-4)*(5*n+2)*(5*n+3)*(5*n+4)*(n+1)/24):
seq(a(n), n=0..40);

G.f.: (4*x^5-24*x^4+256*x^3+1461*x^2+1386*x+42)*x/(x-1)^6.

a(n) = (5*n-4)*(5*n+2)*(5*n+3)*(5*n+4)*(n+1)/24 for n>0, a(0) = 0.

A215546 Number of standard Young tableaux of shape [6n,6].

Original entry on oeis.org

0, 132, 9996, 92092, 451269, 1570800, 4395118, 10559208, 22664655, 44602348, 81921840, 142247364, 235740505, 375609528, 578665362, 865924240, 1263256995, 1802085012, 2520122836, 3462167436, 4680934125, 6237939136, 8204428854, 10662355704, 13705400695

Offset: 0

Alois P. Heinz, Aug 16 2012

Also the number of binary words with 6n 1's and 6 0's such that for every prefix the number of 1's is >= the number of 0's.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Young tableau.

Row n=6 of A214776.

a:= n-> max(0, (6*n-5)*(6*n+5)*(3*n+2)*(2*n+1)*(3*n+1)*(n+1)/10):
seq(a(n), n=0..40);

G.f.: (5*x^6-35*x^5-609*x^4-11921*x^3-24892*x^2-9072*x-132)*x/(x-1)^7.
a(n) = (6*n-5)*(6*n+5)*(3*n+2)*(2*n+1)*(3*n+1)*(n+1)/10 for n>0, a(0) = 0.
Sum_{n>=1} 1/a(n) = 76/385 - 1559*Pi/(924*sqrt(3)) + 2080*log(2)/231 - 135*log(3)/44. - Amiram Eldar, Aug 29 2025

A215547 Number of standard Young tableaux of shape [7n,7].

Original entry on oeis.org

0, 429, 62016, 807300, 5101360, 21732542, 71916768, 199448964, 485325150, 1067658735, 2167714560, 4122884232, 7427426292, 12781794760, 21151379600, 33835482648, 52547352546, 79506102225, 117541332480, 170211285180, 241935349656, 338141745810, 465431207488

Offset: 0

Alois P. Heinz, Aug 16 2012

nonn

Also the number of binary words with 7n 1's and 7 0's such that for every prefix the number of 1's is >= the number of 0's.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Young tableau

Row n=7 of A214776.

a:= n-> max(0, binomial(7*n+7,7)*(7*n-6)/(7*n+1)):
seq(a(n), n=0..30);

Join[{0},Table[Binomial[7n+7,7] (7n-6)/(7n+1),{n,30}]] (* Harvey P. Dale, Jul 24 2016 *)

G.f.: (6*x^7 -48*x^6 +2808*x^5 +83196*x^4 +355384*x^3 +323184*x^2 +58584*x +429)*x / (x-1)^8.

a(n) = C(7*n+7,7)*(7*n-6)/(7*n+1) for n>0, a(0) = 0.

A215548 Number of standard Young tableaux of shape [8n,8].

Original entry on oeis.org

0, 1430, 389367, 7152444, 58261125, 303719922, 1188576675, 3804949176, 10495906641, 25810820750, 57928578191, 120681823860, 236332181085, 439263172458, 780774342075, 1335176857200, 2207407644585, 3542395893894, 5536432537895, 8450810096300, 12628017047349

Offset: 0

Alois P. Heinz, Aug 16 2012

nonn

Also the number of binary words with 8n 1's and 8 0's such that for every prefix the number of 1's is >= the number of 0's.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Young tableau

Row n=8 of A214776.

a:= n-> max(0, binomial(8*n+8,8)*(8*n-7)/(8*n+1)):
seq(a(n), n=0..30);

Join[{0},Table[(Binomial[8n+8,8](8n-7))/(8n+1),{n,20}]] (* Harvey P. Dale, Mar 17 2020 *)

G.f.: (7*x^8 -63*x^7 -9615*x^6 -572643*x^5 -4331133*x^4 -7786221*x^3 -3699621*x^2 -376497*x -1430)*x / (x-1)^9.

a(n) = C(8*n+8,8)*(8*n-7)/(8*n+1) for n>0, a(0) = 0.

cp's OEIS Frontend

A174687 Central coefficients T(2n,n) of the Catalan triangle A033184.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A126596 a(n) = binomial(4*n,n)*(2*n+1)/(3*n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Formula

Extensions

A215541 a(n) = binomial(5*n,n)*(3*n+1)/(4*n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A215542 a(n) = binomial(6*n,n)*(4*n+1)/(5*n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A215543 Number of standard Young tableaux of shape [3n,3].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A215544 Number of standard Young tableaux of shape [4n,4].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A215545 Number of standard Young tableaux of shape [5n,5].

Views

Author

Keywords

Comments

A126596 a(n) = binomial(4n,n)(2n+1)/(3n+1).

A215541 a(n) = binomial(5n,n)(3n+1)/(4n+1).

A215542 a(n) = binomial(6n,n)(4n+1)/(5n+1).