cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A235536 a(n) = binomial(8*n, 2*n) / (6*n + 1).

Original entry on oeis.org

1, 4, 140, 7084, 420732, 27343888, 1882933364, 134993766600, 9969937491420, 753310723010608, 57956002331347120, 4524678117939182220, 357557785658996609700, 28545588568201512137904, 2298872717007844035521848, 186533392975795702301759056
Offset: 0

Views

Author

Bruno Berselli, Jan 12 2014

Keywords

Comments

This is the case l=6, k=2 of binomial((l+k)*n,k*n)/((l*n+1)/gcd(k,l*n+1)), see Theorem 1.1 in Zhi-Wei Sun's paper.
First bisection of A002293.
Also, the sequence is between A002296 and A235535.

Links

Crossrefs

Cf. similar sequences generated by binomial((l+k)*n,k*n)/(l*n+1), where l is divisible by all the factors of k: A000108 (l=1, k=1), A001764 (l=2, k=1), A002293 (l=3, k=1), A002294 (l=4, k=1), A002295 (l=5, k=1), A002296 (l=6, k=1), A007556 (l=7, k=1), A062994 (l=8, k=1), A059968 (l=9, k=1), A230388 (l=10, k=1), A048990 (l=2, k=2), A235534 (l=4, k=2), this sequence (l=6, k=2), A187357 (l=3, k=3), A235535 (l=6, k=3).

Programs

  • Magma
    l:=6; k:=2; [Binomial((l+k)*n,k*n)/(l*n+1): n in [0..20]]; /* where l is divisible by all the prime factors of k */
  • Mathematica
    Table[Binomial[8 n, 2 n]/(6 n + 1), {n, 0, 20}]

Formula

a(n) = A124753(6*n).
From Ilya Gutkovskiy, Jun 21 2018: (Start)
G.f.: 6F5(1/8,1/4,3/8,5/8,3/4,7/8; 1/3,1/2,2/3,5/6,7/6; 65536*x/729).
a(n) ~ 2^(16*n-1)/(sqrt(Pi)*3^(6*n+3/2)*n^(3/2)). (End)

A137211 Generalized or s-Catalan numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 5, 12, 22, 1, 14, 55, 140, 285, 1, 42, 273, 969, 2530, 5481, 1, 132, 1428, 7084, 23751, 62832, 141778, 1, 429, 7752, 53820, 231880, 749398, 1997688, 4638348, 1, 1430, 43263, 420732, 2330445, 9203634, 28989675, 77652024
Offset: 1

Views

Author

Roger L. Bagula, Mar 05 2008

Keywords

Comments

From R. J. Mathar, May 04 2008: (Start)
This is a triangular section of Stanica's array of s-Catalan numbers, with rows A000108, A001764, A002293-A002296, A007556, A062994, A059968,... read along diagonals in A062993 and A070914:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ...
1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, ...
1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, ...
1, 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, ...
1, 1, 6, 51, 506, 5481, 62832, 749398, 9203634, 115607310, 1478314266, ...
1, 1, 7, 70, 819, 10472, 141778, 1997688, 28989675, 430321633, 6503352856, ...
1, 1, 8, 92, 1240, 18278, 285384, 4638348, 77652024, 1329890705, 23190029720, ...
1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, 3573805950, 70625252863, ...
1, 1, 10, 145, 2470, 46060, 910252, 18730855, 397089550, 8612835715, 190223180840, ...
(End)
The Fuss-Catalan numbers are Cat(d,k)= [1/(k*(d-1)+1)]*binomial(k*d,k) and enumerate the number of (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Whieldon and Schuetz link for this interpretation and others), so the (k+1)-th column of Stanica's array enumerates the number of (n+1)-gon partitions of a (k*(n-1)+2)-gon. Cf. A000326 (k=3), A100157 (k=4) and A234043 (k=5). - Tom Copeland, Oct 05 2014

Examples

			{1},
{1, 1},
{1, 2, 3},
{1, 5, 12, 22},
{1, 14, 55, 140, 285},
{1, 42, 273, 969, 2530, 5481},
{1, 132, 1428, 7084, 23751, 62832, 141778},
{1, 429, 7752, 53820, 231880, 749398, 1997688, 4638348}

Links

Programs

  • Mathematica
    t[n_, m_] := Binomial[m*n, n]/((m - 1)*n + 1); a = Table[Table[t[n, m], {m, 1, n + 1}], {n, 0, 10}]; Flatten[a]

Formula

T(n,m) = binomial(m*n,n)/((m-1)*n+1).

Extensions

Edited by N. J. A. Sloane, May 16 2008
Previous Showing 21-22 of 22 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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