A355262 -id:A355262 - cp's OEIS frontend

A355172 The Fuss-Catalan triangle of order 2, read by rows. Related to ternary trees.

1, 0, 1, 0, 1, 3, 0, 1, 5, 12, 0, 1, 7, 25, 55, 0, 1, 9, 42, 130, 273, 0, 1, 11, 63, 245, 700, 1428, 0, 1, 13, 88, 408, 1428, 3876, 7752, 0, 1, 15, 117, 627, 2565, 8379, 21945, 43263, 0, 1, 17, 150, 910, 4235, 15939, 49588, 126500, 246675

Offset: 0

Peter Luschny, Jun 25 2022

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 = 2. (See the Python program for a reference implementation.)

This definition also includes the Fuss-Catalan numbers A001764(n) = T(n, n), or row 3 in A355262. For m = 1 see A355173 and for m = 3 A355174. 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.

			Table T(n, k) begins:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1,  3]
  [3] [0, 1,  5, 12]
  [4] [0, 1,  7, 25,  55]
  [5] [0, 1,  9, 42, 130,  273]
  [6] [0, 1, 11, 63, 245,  700, 1428]
  [7] [0, 1, 13, 88, 408, 1428, 3876, 7752]
Seen as an array reading the diagonals starting from the main diagonal:
  [0] 1, 1,  3, 12,  55,  273,  1428,   7752,   43263,  246675, ...  A001764
  [1] 0, 1,  5, 25, 130,  700,  3876,  21945,  126500,  740025, ...  A102893
  [2] 0, 1,  7, 42, 245, 1428,  8379,  49588,  296010, 1781325, ...  A102594
  [3] 0, 1,  9, 63, 408, 2565, 15939,  98670,  610740, 3786588, ...  A230547
  [4] 0, 1, 11, 88, 627, 4235, 27830, 180180, 1157013, 7396972, ...

A001764 (main diagonal), A102893 (subdiagonal), A102594 (diagonal 2), A230547 (diagonal 3), A005408 (column 2), A071355 (column 3), A006629 (row sums), A143603 (variant).

Cf. A123110 (triangle of order 0), A355173 (triangle of order 1), A355174 (triangle of order 3), A355262 (Fuss-Catalan array).

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(row)))
for n in range(9): print(Trow(n))

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, 2).
T(n, k) = (2*n - 2*k + 3)*(2*n + k - 1)!/((2*n + 1)!*(k - 1)!) for k > 0; T(n, 0) = 0^n.
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 - 3*x^2)/(1 - x)^(2*n + 2)).

A355173 The Fuss-Catalan triangle of order 1, read by rows. Related to binary trees.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 4, 9, 14, 0, 1, 5, 14, 28, 42, 0, 1, 6, 20, 48, 90, 132, 0, 1, 7, 27, 75, 165, 297, 429, 0, 1, 8, 35, 110, 275, 572, 1001, 1430, 0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 0, 1, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796

Offset: 0

Peter Luschny, Jun 25 2022

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 = 1. (See the Python program for a reference implementation.)

This definition also includes the classical Fuss-Catalan numbers, since T(n, n) = A000108(n), or row 2 in A355262. For m = 2 see A355172 and for m = 3 A355174. 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.

			Table T(n, k) begins:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1, 2]
  [3] [0, 1, 3,  5]
  [4] [0, 1, 4,  9,  14]
  [5] [0, 1, 5, 14,  28,  42]
  [6] [0, 1, 6, 20,  48,  90,  132]
  [7] [0, 1, 7, 27,  75, 165,  297, 429]
  [8] [0, 1, 8, 35, 110, 275,  572, 1001, 1430]
  [9] [0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862]
Seen as an array reading the diagonals starting from the main diagonal:
  [0] 1, 1, 2,  5,  14,   42,  132,   429,  1430,   4862,   16796, ...  A000108
  [1] 0, 1, 3,  9,  28,   90,  297,  1001,  3432,  11934,   41990, ...  A000245
  [2] 0, 1, 4, 14,  48,  165,  572,  2002,  7072,  25194,   90440, ...  A099376
  [3] 0, 1, 5, 20,  75,  275, 1001,  3640, 13260,  48450,  177650, ...  A115144
  [4] 0, 1, 6, 27, 110,  429, 1638,  6188, 23256,  87210,  326876, ...  A115145
  [5] 0, 1, 7, 35, 154,  637, 2548,  9996, 38760, 149226,  572033, ...  A000588
  [6] 0, 1, 8, 44, 208,  910, 3808, 15504, 62016, 245157,  961400, ...  A115147
  [7] 0, 1, 9, 54, 273, 1260, 5508, 23256, 95931, 389367, 1562275, ...  A115148

A000108 (main diagonal), A000245 (subdiagonal), A002057 (diagonal 2), A000344 (diagonal 3), A000027 (column 2), A000096 (column 3), A071724 (row sums), A000958 (alternating row sums), A262394 (main diagonal of array).
Variants: A009766 (main variant), A030237, A130020.
Cf. A123110 (triangle of order 0), A355172 (triangle of order 2), A355174 (triangle of order 3), A355262 (Fuss-Catalan array).
Cf. A115144, A115145, A000588, A115147, A115148.

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(row))
for n in range(11): print(Trow(n))

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, 1).
T(n, k) = (n - k + 2)*(n + k - 1)!/((n + 1)!*(k - 1)!) for k > 0; T(n, 0) = 0^n.
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 - 2*x^2)/(1 - x)^(n + 2)).

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

Peter Luschny, Jun 25 2022

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.

			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, ...

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).

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))

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)).

A362214 a(n) = the hypergraph Fuss-Catalan number FC_(2,2)(n).

Original entry on oeis.org

1, 1, 144, 1341648, 693520980336

Offset: 0

Peter Bala, Apr 11 2023

Chavan et al. associate to each pair (r,m) of positive integers the sequence of hypergraph Fuss-Catalan numbers {FC_(r,m)(n) : n >= 0}. This is the case (r,m) = (2,2).
When m = 1, the sequence {FC_(r,1)(n) : n >= 0} is equivalent to the sequence of Fuss-Catalan numbers { (1/(r*n+1))*binomial((r+1)*n,n) : n >= 0}. Note that r = 1 corresponds to the Catalan numbers A000108. See A355262 for a table of Fuss-Catalan numbers.
When r = 1, the sequence {FC_(1,m)(n) : n >= 0} is equivalent to the sequence of hypergraph Catalan numbers {C_m(n) : n >= 0}. See A362167 - A362172 for the cases m = 2 through 7.

Parth Chavan, Andrew Lee and Karthik Seetharaman, Hypergraph Fuss-Catalan numbers, arXiv:2202.01111 [math.CO], 2022.

Cf. A000108, A355262, A362167, A362215, A362216, A362217.

A362215 a(n) = the hypergraph Fuss-Catalan number FC_(2,3)(n).

Original entry on oeis.org

1, 1, 480, 200225, 18527520, 45589896150400

Offset: 0

Peter Bala, Apr 11 2023

Chavan et al. associate to each pair (r,m) of positive integers the sequence of hypergraph Fuss-Catalan numbers {FC_(r,m)(n) : n >= 0}. This is the case (r,m) = (2,3).
When m = 1, the sequence {FC_(r,1)(n) : n >= 0} is equivalent to the sequence of Fuss-Catalan numbers { (1/(r*n+1))*binomial((r+1)*n,n) : n >= 0}. Note that r = 1 corresponds to the Catalan numbers A000108. See A355262 for a table of Fuss-Catalan numbers.
When r = 1, the sequence {FC_(1,m)(n) : n >= 0} is equivalent to the sequence of hypergraph Catalan numbers {C_m(n) : n >= 0}. See A362167 - A362172 for the cases m = 2 through 7.

Parth Chavan, Andrew Lee and Karthik Seetharaman, Hypergraph Fuss-Catalan numbers, arXiv:2202.01111 [math.CO], 2022.

Cf. A000108, A355262, A362167, A362214, A362216, A362217.

A362216 a(n) = the hypergraph Fuss-Catalan number FC_(3,2)(n).

Original entry on oeis.org

1, 1, 11532, 628958939250, 163980917165716725552156

Offset: 0

Peter Bala, Apr 11 2023

Chavan et al. associate to each pair (r,m) of positive integers the sequence of hypergraph Fuss-Catalan numbers {FC_(r,m)(n) : n >= 0}. This is the case (r,m) = (3,2).
When m = 1, the sequence {FC_(r,1)(n) : n >= 0} is equivalent to the sequence of Fuss-Catalan numbers { (1/(r*n+1))*binomial((r+1)*n,n) : n >= 0}. Note that r = 1 corresponds to the Catalan numbers A000108. See A355262 for a table of Fuss-Catalan numbers.
When r = 1, the sequence {FC_(1,m)(n) : n >= 0} is equivalent to the sequence of hypergraph Catalan numbers {C_m(n) : n >= 0}. See A362167 - A362172 for the cases m = 2 through 7.

Parth Chavan, Andrew Lee and Karthik Seetharaman, Hypergraph Fuss-Catalan numbers, arXiv:2202.01111 [math.CO], 2022.

Cf. A000108, A355262, A362167, A362214, A362215, A362217.

A362217 a(n) = the hypergraph Fuss-Catalan number FC_(3,3)(n).

Original entry on oeis.org

1, 1, 38440, 8272793255000, 9396808005460764741084000

Offset: 0

Peter Bala, Apr 11 2023

Chavan et al. associate to each pair (r,m) of positive integers the sequence of hypergraph Fuss-Catalan numbers {FC_(r,m)(n) : n >= 0}. This is the case (r,m) = (3,3).
When m = 1, the sequence {FC_(r,1)(n) : n >= 0} is equivalent to the sequence of Fuss-Catalan numbers { (1/(r*n+1))*binomial((r+1)*n,n) : n >= 0}. Note that r = 1 corresponds to the Catalan numbers A000108. See A355262 for a table of Fuss-Catalan numbers.
When r = 1, the sequence {FC_(1,m)(n) : n >= 0} is equivalent to the sequence of hypergraph Catalan numbers {C_m(n) : n >= 0}. See A362167 - A362172 for the cases m = 2 through 7.

Parth Chavan, Andrew Lee and Karthik Seetharaman, Hypergraph Fuss-Catalan numbers, arXiv:2202.01111 [math.CO], 2022.

Cf. A000108, A355262, A362167, A362214, A362215, A362216.

A382100 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of 1/(2 - B_k(x)), where B_k(x) = 1 + x*B_k(x)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 10, 8, 1, 1, 1, 5, 19, 35, 16, 1, 1, 1, 6, 31, 98, 126, 32, 1, 1, 1, 7, 46, 213, 531, 462, 64, 1, 1, 1, 8, 64, 396, 1556, 2974, 1716, 128, 1, 1, 1, 9, 85, 663, 3651, 11843, 17060, 6435, 256, 1

Offset: 0

Seiichi Manyama, Mar 15 2025

			Square array begins:
  1,  1,   1,    1,     1,     1,     1, ...
  1,  1,   1,    1,     1,     1,     1, ...
  1,  2,   3,    4,     5,     6,     7, ...
  1,  4,  10,   19,    31,    46,    64, ...
  1,  8,  35,   98,   213,   396,   663, ...
  1, 16, 126,  531,  1556,  3651,  7391, ...
  1, 32, 462, 2974, 11843, 35232, 86488, ...

Columns k=0..5 give A000012, A011782, A088218, A047099 (for n > 0), A107026(n)/2 (for n > 0), A304979(n)/2 (for n > 0).

Cf. A107027, A107030, A355262, A382101.

a(n, k) = polcoef(1/(2-sum(j=0, n, binomial(k*j+1, j)/(k*j+1)*x^j+x*O(x^n))), n);

a(n, k) = if(n==0, 1, (binomial(k*n, n)-(k-2)*sum(j=0, n-1, binomial(k*n, j)))/2);

A(n,k) = ( binomial(k*n,n) - (k-2) * Sum_{j=0..n-1} binomial(k*n,j) )/2 for n > 0.

G.f. of column k: 1/( 1 - Series_Reversion( x/(1+x)^k ) ).

A382101 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(B_k(x) - 1), where B_k(x) = 1 + x*B_k(x)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 43, 73, 1, 1, 1, 9, 91, 529, 501, 1, 1, 1, 11, 157, 1753, 8501, 4051, 1, 1, 1, 13, 241, 4129, 45001, 169021, 37633, 1, 1, 1, 15, 343, 8041, 146001, 1447471, 4010455, 394353, 1, 1, 1, 17, 463, 13873, 362501, 6502681, 56041987, 110676833, 4596553, 1

Offset: 0

Seiichi Manyama, Mar 15 2025

			Square array begins:
  1,   1,    1,     1,      1,      1, ...
  1,   1,    1,     1,      1,      1, ...
  1,   3,    5,     7,      9,     11, ...
  1,  13,   43,    91,    157,    241, ...
  1,  73,  529,  1753,   4129,   8041, ...
  1, 501, 8501, 45001, 146001, 362501, ...

Columns k=0..4 give A000012, A000262, A251568, A380512, A380516.

Cf. A355262, A382100.

a(n, k) = if(n==0, 1, (n-1)!*pollaguerre(n-1, (k-1)*n+1, -1));

A(n,k) = (n-1)! * Sum_{j=0..n-1} binomial(k*n,j)/(n-j-1)! for n > 0.
A(n,k) = (n-1)! * LaguerreL(n-1, (k-1)*n+1, -1) for n > 0.
E.g.f. of column k: exp( Series_Reversion( x/(1+x)^k ) ).

cp's OEIS Frontend

A355172 The Fuss-Catalan triangle of order 2, read by rows. Related to ternary trees.

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A355173 The Fuss-Catalan triangle of order 1, read by rows. Related to binary trees.

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

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A362214 a(n) = the hypergraph Fuss-Catalan number FC_(2,2)(n).

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A362215 a(n) = the hypergraph Fuss-Catalan number FC_(2,3)(n).

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A362216 a(n) = the hypergraph Fuss-Catalan number FC_(3,2)(n).

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A362217 a(n) = the hypergraph Fuss-Catalan number FC_(3,3)(n).

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A382100 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of 1/(2 - B_k(x)), where B_k(x) = 1 + x*B_k(x)^k.

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A382101 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(B_k(x) - 1), where B_k(x) = 1 + x*B_k(x)^k.

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