A306444 -id:A306444 - cp's OEIS frontend

A265101 a(n) = binomial(6n + 5, 3n + 1)/(6*n + 5).

1, 30, 1144, 49742, 2340135, 115997970, 5967382200, 315614844558, 17055399281284, 937581428480312, 52267355178398304, 2947837630317717410, 167897169647656366330, 9643503773422181941740, 557939244828083793388560, 32486374828326106197187470

Offset: 0

Peter Bala, Dec 02 2015

Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = 1/(2*p + q)*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n + 1/3).

Number of maximal faces of the rational associahedron Ass(3*n + 1, 3*n + 4). Number of lattice paths from (0, 0) to (3*n + 4, 3*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (3*n + 1)/(3*n + 4)*x. See Armstrong et al.

Seiichi Manyama, Table of n, a(n) for n = 0..555
D. Armstrong, B. Rhoades, and N. Williams, Rational associahedra and noncrossing partitions arxiv:1305.7286v1 [math.CO], 2013.

Row 3 of A306444.

Cf. A000108, A065097 (Cat(n + 1/2)), A265102 (Cat(n + 1/4)), A265103 (Cat(n + 1/5)).

[Binomial(6*n+5, 3*n+1)/(6*n+5): n in [0..15]]; // Vincenzo Librandi, Dec 09 2015

seq(1/(6*n + 5)*binomial(6*n + 5, 3*n + 1), n = 0..15);

Table[1/(6 n + 5) Binomial[6 n + 5, 3 n + 1], {n, 0, 20}] (* Vincenzo Librandi, Dec 09 2015 *)

a(n) = binomial(6*n + 5, 3*n + 1)/(6*n + 5); \\ Altug Alkan, Dec 07 2015

[binomial(6*n+5, 3*n+1)/(6*n+5) for n in (0..15)] # G. C. Greubel, Feb 16 2019

a(n) = binomial(6*n + 5, 3*n + 1)/(6*n + 5).
(n + 1)*(3*n - 1)*(3*n + 4)*a(n) = 8*(2*n + 1)*(6*n + 1)*(6*n - 1)*a(n-1) with a(0) = 1.
From Ilya Gutkovskiy, Feb 28 2017: (Start)
O.g.f.: (3F2(-1/6,1/6,1/2; -1/3,4/3; 64*x) - 1)/(2*x).
E.g.f.: 3F3(5/6,7/6,3/2; 2/3,2,7/3; 64*x).
a(n) ~ 4^(3*n+2)/(3*sqrt(3*Pi)*n^(3/2)). (End)

A265102 a(n) = binomial(8n + 6, 4n + 1)/(8*n + 6).

Original entry on oeis.org

1, 143, 22610, 3991995, 757398510, 150946230006, 31170212479588, 6611198199648595, 1431806849011462742, 315319074704135127010, 70398290295706497441660, 15897587681946817926283230, 3624898901185998294920196300, 833406923656808938891174678092

Offset: 0

Peter Bala, Dec 02 2015

Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = 1/(2*p + q)*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n + 1/4).

Number of maximal faces of the rational associahedron Ass(4*n + 1, 4*n + 5). Number of lattice paths from (0, 0) to (4*n + 5, 4*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (4*n + 1)/(4*n + 5)*x. See Armstrong et al.

Seiichi Manyama, Table of n, a(n) for n = 0..416
D. Armstrong, B. Rhoades, and N. Williams, Rational associahedra and noncrossing partitions arxiv:1305.7286v1 [math.CO], 2013.

Row 4 of A306444.

Cf. A000108, A065097 (Cat(n + 1/2)), A265101 (Cat(n + 1/3)), A265103 (Cat(n + 1/5)).

[Binomial(8*n+6, 4*n+1)/(8*n+6): n in [0..20]]; // G. C. Greubel, Feb 16 2019

seq(binomial(8*n + 6, 4*n + 1)/(8*n + 6), n = 0..14);

Table[Binomial[8n+6, 4n+1]/(8n+6), {n, 0, 20}] (* Vincenzo Librandi, Dec 09 2015 *)

a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6); \\ Altug Alkan, Dec 07 2015

[binomial(8*n+6, 4*n+1)/(8*n+6) for n in (0..20)] # G. C. Greubel, Feb 16 2019

a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6).
(n + 1)*(2*n - 1)*(4*n + 3)*(4*n + 5)*a(n) = 2*(8*n + 1)*(8*n - 1)*(8*n + 3)*(8*n + 5)*a(n-1) with a(0) = 1.
From Ilya Gutkovskiy, Feb 28 2017: (Start)
O.g.f.: (4F3(-1/8,1/8,3/8,5/8; -1/2,3/4,5/4; 256*x) - 1)/(2*x).
E.g.f.: 4F4(7/8,9/8,11/8,13/8; 1/2,7/4,2,9/4; 256*x).
a(n) ~ 4^(4*n+1)/(sqrt(Pi)*n^(3/2)). (End)

A265103 a(n) = binomial(10n + 7, 5n + 1)/(10*n + 7).

Original entry on oeis.org

1, 728, 482885, 347993910, 267058714626, 214401560777712, 177957899774070416, 151516957974714281810, 131614194900668669130060, 116186564091895720987588128, 103938666796148178180041038716, 94020887900502277905668153549928, 85855382816448334044679630209920925

Offset: 0

Peter Bala, Dec 02 2015

Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = 1/(2*p + q)*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n + 1/5).

Number of maximal faces of the rational associahedron Ass(5*n + 1, 5*n + 6). Number of lattice paths from (0, 0) to (5*n + 6, 5*n + 1) using steps of the form (1, 0) and (0, 1) and staying above the line y = (5*n + 1)/(5*n + 6)*x. See Armstrong et al.

Seiichi Manyama, Table of n, a(n) for n = 0..333
D. Armstrong, B. Rhoades, and N. Williams, Rational associahedra and noncrossing partitions arxiv:1305.7286v1 [math.CO], 2013.

Row 5 of A306444.

Cf. A000108, A065097 (Cat(n + 1/2)), A265101 (Cat(n + 1/3)), A265102 (Cat(n + 1/4)).

[Binomial(10*n+7, 5*n+1)/(10*n+7): n in [0..15]]; // Vincenzo Librandi, Dec 09 2015

seq(binomial(10*n + 7, 5*n + 1)/(10*n + 7), n = 0..12);

Table[Binomial[10n+7, 5n+1]/(10n+7), {n, 0, 20}] (* Vincenzo Librandi, Dec 09 2015 *)

a(n)=binomial(10*n + 7, 5*n + 1)/(10*n + 7) \\ Anders Hellström, Dec 07 2015

[binomial(10*n+7, 5*n+1)/(10*n+7) for n in (0..20)] # G. C. Greubel, Feb 16 2019

a(n) = binomial(10*n + 7, 5*n + 1)/(10*n + 7).
(n + 1)*(5*n - 2)*(5*n - 3)*(5*n + 4)*(5*n + 6)*a(n) = 32*(2*n + 1)*(10*n + 1)*(10*n - 1)*(10*n + 3)*(10*n - 3)*a(n-1) with a(0) = 1.
From Ilya Gutkovskiy, Feb 28 2017: (Start)
O.g.f.: (5F4(-3/10,-1/10,1/10,3/10,1/2; -3/5,-2/5,4/5,6/5; 1024*x) - 1)/(2*x).
E.g.f.: 5F5(7/10,9/10,11/10,13/10,3/2; 2/5,3/5,9/5,2,11/5; 1024*x).
a(n) ~ 4^(5*n+3)/(5*sqrt(5*Pi)*n^(3/2)). (End)

cp's OEIS Frontend

A265101 a(n) = binomial(6*n + 5, 3*n + 1)/(6*n + 5).

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A265102 a(n) = binomial(8*n + 6, 4*n + 1)/(8*n + 6).

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Author

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Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A265103 a(n) = binomial(10*n + 7, 5*n + 1)/(10*n + 7).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A265101 a(n) = binomial(6n + 5, 3n + 1)/(6*n + 5).

A265102 a(n) = binomial(8n + 6, 4n + 1)/(8*n + 6).

A265103 a(n) = binomial(10n + 7, 5n + 1)/(10*n + 7).