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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A006635 a(n) = 4*binomial(4*n+11, n)/(n+4).

Original entry on oeis.org

1, 12, 114, 1012, 8775, 75516, 649264, 5593068, 48336171, 419276660, 3650774820, 31907617560, 279871768995, 2463161027292, 21747225841440, 192575673551584, 1710009515037060, 15223466050169520, 135853465827080970, 1215067013768834100, 10890252031152078585
Offset: 0

Views

Author

Simon Plouffe

Keywords

Comments

Former name: From generalized Catalan numbers. - G. C. Greubel, Sep 02 2025

References

  • H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002293, A196678, A006633, A233658, A233666, A006634, A233667.

Programs

Formula

G.f.: 4F3([3,7/2,15/4,13/4], [5,14/3,13/3], 256*x/27). - Simon Plouffe, Master's thesis, UQAM 1992
G.f.: g^12 where g is the g.f. of A002293. - Sean A. Irvine, May 25 2017
a(n) = 4*binomial(4*n+11, n)/(n+4). - G. C. Greubel, Sep 02 2025

Extensions

More terms from Sean A. Irvine, May 25 2017
New name by G. C. Greubel, Sep 02 2025

A355174 The Fuss-Catalan triangle of order 3, read by rows. Related to quartic trees.

Original entry on oeis.org

1, 0, 1, 0, 1, 4, 0, 1, 7, 22, 0, 1, 10, 49, 140, 0, 1, 13, 85, 357, 969, 0, 1, 16, 130, 700, 2695, 7084, 0, 1, 19, 184, 1196, 5750, 20930, 53820, 0, 1, 22, 247, 1872, 10647, 47502, 166257, 420732, 0, 1, 25, 319, 2755, 17980, 93496, 395560, 1344904, 3362260
Offset: 0

Views

Author

Peter Luschny, Jun 25 2022

Keywords

Comments

The Fuss-Catalan triangle of order m is a regular, (0, 0)-based table recursively defined as follows: Set row(0) = [1] and row(1) = [0, 1]. Now assume row(n-1) already constructed and duplicate the last element of row(n-1). Next apply the cumulative sum m times to this list to get row(n). Here m = 3. (See the Python program for a reference implementation.)
This definition also includes the Fuss-Catalan numbers A002293(n) = T(n, n), row 4 in A355262. For m = 1 see A355173 and for m = 2 A355172. More generally, for n >= 1 all Fuss-Catalan sequences (A355262(n, k), k >= 0) are the main diagonals of the Fuss-Catalan triangles of order n - 1.

Examples

			Table T(n, k) begins:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1,  4]
  [3] [0, 1,  7,  22]
  [4] [0, 1, 10,  49,  140]
  [5] [0, 1, 13,  85,  357,   969]
  [6] [0, 1, 16, 130,  700,  2695,  7084]
  [7] [0, 1, 19, 184, 1196,  5750, 20930,  53820]
  [8] [0, 1, 22, 247, 1872, 10647, 47502, 166257,  420732]
  [9] [0, 1, 25, 319, 2755, 17980, 93496, 395560, 1344904, 3362260]
Seen as an array reading the diagonals starting from the main diagonal:
  [0] 1, 1,  4,  22,  140,   969,   7084,   53820,   420732, ...  A002293
  [1] 0, 1,  7,  49,  357,  2695,  20930,  166257,  1344904, ...  A233658
  [2] 0, 1, 10,  85,  700,  5750,  47502,  395560,  3321120, ...  A233667
  [3] 0, 1, 13, 130, 1196, 10647,  93496,  816816,  7128420, ...
  [4] 0, 1, 16, 184, 1872, 17980, 167552, 1535352, 13934752, ...

Crossrefs

A002293 (main diagonal), A233658 (subdiagonal), A233667 (diagonal 2), A016777 (column 2), A196678 (row sums).
Cf. A123110 (triangle of order 0), A355173 (triangle of order 1), A355172 (triangle of order 2), A355262 (Fuss-Catalan array).

Programs

  • Python
    from functools import cache
from itertools import accumulate
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 1]
    row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
    return list(accumulate(accumulate(accumulate(row))))
for n in range(11): print(Trow(n))

Formula

The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
FCT(n, k, m) = (m*(n - k) + m + 1)*(m*n + k - 1)!/((m*n + 1)!*(k - 1)!) for k > 0 and FCT(n, 0, m) = 0^n. The case considered here is T(n, k) = FCT(n, k, 3).
T(n, k) = (3*(n - k) + 4)*(3*n + k - 1)!/((3*n + 1)!*(k - 1)!) for k > 0; T(n, 0) = n^0.
The g.f. of row n of the FC-triangle of order m is 0^n + (x - (m + 1)*x^2) / (1 - x)^(m*n + 2), thus:
T(n, k) = [x^k] (0^n + (x - 4*x^2)/(1 - x)^(3*n + 2)).

A268315 Decimal expansion of 256/27.

Original entry on oeis.org

9, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8
Offset: 1

Views

Author

Gheorghe Coserea, Feb 01 2016

Keywords

Examples

			9.481481481481481481481481481481...

Crossrefs

Exponential growth rate for A000260, A002293, A005810, A006633, A006634, A052203, A066381, A069271, A078516, A078995, A153396, A196678, A268316.

Programs

  • Magma
    [9] cat &cat[[4, 8, 1]^^45]; // Vincenzo Librandi, Feb 04 2016
  • Mathematica
    Join[{9}, PadRight[{}, 120, {4, 8, 1}]] (* Vincenzo Librandi, Feb 04 2016 *)
  • PARI
    1.0 * 256/27

Extensions

More digits from Jon E. Schoenfield, Mar 15 2018
Previous Showing 11-13 of 13 results.

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