A060540 -id:A060540 - cp's OEIS frontend

A369288 Array read by antidiagonals: A(n,k) = the hypergraph Catalan number C_k(n), n >= 0, k >= 1.

1, 1, 1, 1, 1, 2, 1, 1, 6, 5, 1, 1, 20, 57, 14, 1, 1, 70, 860, 678, 42, 1, 1, 252, 15225, 57200, 9270, 132, 1, 1, 924, 299880, 7043750, 5344800, 139968, 429, 1, 1, 3432, 6358044, 1112865264, 6327749750, 682612800, 2285073, 1430, 1, 1, 12870, 141858288, 203356067376, 11126161436292, 10411817136000, 118180104000, 39871926, 4862

Offset: 0

Andrew Howroyd, Feb 01 2024

Definition (from A362167): Let Trees(n) be the set of unlabeled trees on n vertices (see A000055). Let T be in Trees(n+1), and let v be a vertex of T. Then a (k,T)-tour beginning at v is a walk that begins and ends at v and traverses each edge of T exactly 2*k times. We denote by N(k,T)(v) the number of (k,T)-tours beginning at v.
The hypergraph Catalan numbers C_k(n) are defined by C_k(n) = Sum_{trees T in T(n+1)} Sum_{vertices v in T} N(k,T)(v)/|Aut(T)|, where Aut(T) denotes the automorphism group of the tree T.
See the Gunnells reference for a full definition and additional information.

			Array begins:
n/k|   1       2            3              4                  5 ...
---+-----------------------------------------------------------------
 0 |   1       1            1              1                  1 ...
 1 |   1       1            1              1                  1 ...
 2 |   2       6           20             70                252 ...
 3 |   5      57          860          15225             299880 ...
 4 |  14     678        57200        7043750         1112865264 ...
 5 |  42    9270      5344800     6327749750     11126161436292 ...
 6 | 132  139968    682612800 10411817136000 255654847841227632 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021.

Columns k=1..7 are A000108, A362167, A362168, A362169, A362170, A362171, A362172.
Row 2 is A000984.
Cf. A000055, A060540.

\\ here L(k,n) is k-th column of A060540 as g.f.
L(k,n)={sum(n=1, n, (n*k)!*x^n/(k!^n*n!), O(x*x^n))}
HypCatColGf(k,n)={my(p=L(k,n)); 1 + subst(p, x, serreverse(x^2/p))}
M(n,m=n+1)={Mat(vector(m, k, Col(HypCatColGf(k,n))))}
{ my(A=M(7,5)); for(i=1, matsize(A)[1], print(A[i,])) }

G.f. of column k: 1 + B_k(Series_Reversion(x^2/B_k(x))) where B_k(x) is the g.f. of column k of A060540.

A370363 Number A(n,k) of partitions of [k*n] into n sets of size k having at least one set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 7, 1, 1, 0, 1, 1, 28, 45, 1, 1, 0, 1, 1, 103, 1063, 401, 1, 1, 0, 1, 1, 376, 22893, 74296, 4355, 1, 1, 0, 1, 1, 1384, 503751, 13080721, 8182855, 56127, 1, 1, 0, 1, 1, 5146, 11432655, 2443061876, 15237712355, 1305232804, 836353, 1, 1

Offset: 0

Alois P. Heinz, Feb 16 2024

			A(3,2) = 7: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 14|23|56, 15|26|34, 16|25|34.
Square array A(n,k) begins:
  0, 0,   0,     0,        0,          0, ...
  1, 1,   1,     1,        1,          1, ...
  1, 1,   1,     1,        1,          1, ...
  1, 1,   7,    28,      103,        376, ...
  1, 1,  45,  1063,    22893,     503751, ...
  1, 1, 401, 74296, 13080721, 2443061876, ...

Alois P. Heinz, Antidiagonals n = 0..55, flattened
Wikipedia, Partition of a set

Columns k=0+1,2-3 give: A057427, A370253, A370358.
Rows n=0,1+2,3 give: A000004, A000012, A370487.
Main diagonal gives A370364.
Antidiagonal sums give A370365.
Cf. A060540, A370366.

A:= proc(n, k) option remember; `if`(k=0, signum(n), add(
      (-1)^(n-j+1)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A(n,k) = A060540(n,k) - A370366(n,k) for n,k >= 1.

A025037 Number of partitions of { 1, 2, ..., 5n } into sets of size 5.

Original entry on oeis.org

1, 1, 126, 126126, 488864376, 5194672859376, 123378675083039376, 5721809435651034101376, 470624547891733205872277376, 63887753000850674430367526069376, 13536281554808237495608549953475109376, 4280862577989659916223699531336456815269376

Offset: 0

David W. Wilson

Vincenzo Librandi, Table of n, a(n) for n = 0..100 (term a(0) added by Sidney Cadot)
Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 17.

Column k=5 of A060540.

[Factorial(5*n)/(Factorial(n)*Factorial(5)^n): n in [0..10]]; // Vincenzo Librandi, Jun 26 2012

Table[(5n)!/(n!(5!)^n),{n,0,10}] (* Vincenzo Librandi, Jun 26 2012 *)

[rising_factorial(n+1,4*n)/120^n for n in (0..15)] # Peter Luschny, Jun 26 2012

a(n) = (5n)!/(n!(5!)^n). - Christian G. Bower, Sep 15 1998

a(n) ~ 5^(4*n+1/2) * (n/e)^(4*n) / 24^n. - Amiram Eldar, Aug 28 2025

a(0) from Peter Luschny, Apr 24 2023

A322252 a(0) = 1 and a(n) = (5n)!/(5!n!^5) for n > 0.

Original entry on oeis.org

1, 1, 945, 1401400, 2546168625, 5194672859376, 11423951396577720, 26478825654361766400, 63805953776276649848625, 158421985022100255941485000, 402789797982510165934296910320, 1044048983553856888083223814102400, 2749848597736878877579660426025283000

Offset: 0

Seiichi Manyama, Nov 30 2018

nonn

Row 5 of A060540.

Cf. A000142, A100734, A008978, A060542, A082368.

[1] cat [Factorial(5*n)/(120*Factorial(n)^5):n in [1..12]]; // Marius A. Burtea, Feb 18 2020

a[n_]:=(5*n)!/(5!*n!^5); Array[a, 20] (* or *) CoefficientList[Series[HypergeometricPFQ[{1/5, 2/5, 3/5, 4/5}, {1, 1, 1}, 3125 x]/(120 x) , {x, 0, 20}], x] (* Stefano Spezia, Dec 01 2018 *)

O.g.f.: F({1/5, 2/5, 3/5, 4/5}, {1, 1, 1}, 3125*x)/(120*x), where F is the generalized hypergeometric function. - Stefano Spezia, Dec 01 2018

a(n) = (1/5!)*A008978(n) for n >= 1. - Peter Bala, Feb 18 2020

A370366 Number A(n,k) of partitions of [k*n] into n sets of size k having no set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 9, 8, 0, 0, 1, 0, 34, 252, 60, 0, 0, 1, 0, 125, 5672, 14337, 544, 0, 0, 1, 0, 461, 125750, 2604732, 1327104, 6040, 0, 0, 1, 0, 1715, 2857472, 488360625, 2533087904, 182407545, 79008, 0, 0

Offset: 0

Alois P. Heinz, Feb 16 2024

			A(2,3) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
Square array A(n,k) begins:
  1, 1,   1,       1,          1,             1, ...
  0, 0,   0,       0,          0,             0, ...
  0, 0,   2,       9,         34,           125, ...
  0, 0,   8,     252,       5672,        125750, ...
  0, 0,  60,   14337,    2604732,     488360625, ...
  0, 0, 544, 1327104, 2533087904, 5192229797500, ...

Alois P. Heinz, Antidiagonals n = 0..54, flattened
Wikipedia, Partition of a set

Columns k=0+1,2-3 give: A000007, A053871, A370357.
Rows n=0-2 give: A000012, A000004, A010763(n-1) for k>0.
Main diagonal gives A370367.
Antidiagonal sums give A370368.
Cf. A060540, A370363.

A:= proc(n, k) `if`(k=0,`if`(n=0, 1, 0), add(
      (-1)^(n-j)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A(n,k) = A060540(n,k) - A370363(n,k) for n,k >= 1.

A370407 Total sum over all j in [n] of the number of partitions of [j*(n-j)] into (n-j) sets of size j.

Original entry on oeis.org

1, 2, 3, 4, 7, 29, 424, 22250, 4166012, 3228619112, 9836415861419, 148021077093705105, 9516162824804128833773, 3369338041967340627557507931, 5792066385997100947453116161699033, 55416753515944143275546728017602371379095

Offset: 0

Alois P. Heinz, Feb 17 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..54

Cf. A060540, A275973, A361948, A370365, A370368.

a:= n-> add(`if`(j=n, 1, (j*(n-j))!/((n-j)!^j*j!)), j=0..n):
seq(a(n), n=0..15);

a(n) = A370365(n) + A370368(n).
a(n) = Sum_{j=0..n} A361948(j,n-j).
a(n) mod 2 = A275973(n-1) for n>=2.

A025038 Number of partitions of { 1, 2, ..., 6n } into sets of size 6.

Original entry on oeis.org

1, 1, 462, 2858856, 96197645544, 11423951396577720, 3708580189773818399040, 2779202577056119960603777920, 4263127221846887596248598498826880, 12233832241625685631640659383106015132800, 61247286460823449786646954166350590676638060800

Offset: 0

David W. Wilson

Andrew Howroyd, Table of n, a(n) for n = 0..50
Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 17.

Column k=6 of A060540.

Table[Pochhammer[n + 1, 5*n]/6!^n, {n, 0, 15}] (* Paolo Xausa, Aug 08 2024 *)

[rising_factorial(n+1,5*n)/720^n for n in (0..15)] # Peter Luschny, Jun 26 2012

a(n) = (6n)!/(n!(6!)^n). - Christian G. Bower, Sep 15 1998

a(n) ~ 2^(2*n+1/2) * 3^(4*n+1/2) * (n/e)^(5*n) / 5^n. - Amiram Eldar, Aug 28 2025

a(0) and a(10) from Andrew Howroyd, Feb 26 2018

A025040 Number of partitions of { 1, 2, ..., 8n } into sets of size 8.

Original entry on oeis.org

1, 1, 6435, 1577585295, 4148378852099625, 63805953776276649848625, 4012852078114749147678149338875, 814318942973348333484015877548157809375, 450538787986875167583433232345723106006796340625, 599167346385710947364525167684682505182168120225201390625

Offset: 0

David W. Wilson

Andrew Howroyd, Table of n, a(n) for n = 0..50
Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018.

Column k=8 of A060540.

Table[Pochhammer[n + 1, 7*n]/8!^n, {n, 0, 10}] (* Paolo Xausa, Aug 08 2024 *)

a(n) = (8n)!/(n!(8!)^n). - Christian G. Bower, Sep 15 1998

a(n) ~ 2^(17*n+3/2) * (n/e)^(7*n) / 315^n. - Amiram Eldar, Aug 28 2025

a(0)=1 from Andrew Howroyd, Feb 26 2018

A025041 Number of partitions of { 1, 2, ..., 9n } into sets of size 9.

Original entry on oeis.org

1, 1, 24310, 37978905250, 893864677761055000, 158421985022100255941485000, 140413003088367308737750586624350000, 474750200441159213998774295008486246570750000, 5050927030108676304976606530597710043478228399030000000

Offset: 0

David W. Wilson

Andrew Howroyd, Table of n, a(n) for n = 0..50
Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018.

Column k=9 of A060540.

Table[Pochhammer[n + 1, 8*n]/9!^n, {n, 0, 10}] (* Paolo Xausa, Aug 08 2024 *)

a(n) = (9*n)!/(n!*(9!)^n). - Christian G. Bower, Sep 15 1998

a(n) ~ 3^(14*n+1) * (n/e)^(8*n) / 4480^n. - Amiram Eldar, Aug 28 2025

a(0)=1 from Andrew Howroyd, Feb 26 2018

A025042 Number of partitions of { 1, 2, ..., 10n } into sets of size 10.

Original entry on oeis.org

1, 1, 92378, 925166131890, 196056702961398759480, 402789797982510165934296910320, 5061324188732823772720935900249118313520, 286835743456312434671347570864365730919777702885760, 59034098562652855324502713832050720577190114808212674462486400

Offset: 0

David W. Wilson

Andrew Howroyd, Table of n, a(n) for n = 0..50
Cyril Banderier, Philippe Marchal, and Michael Wallner, Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version), arXiv:1805.09017 [cs.DM], 2018.

Column k=10 of A060540.

Table[Pochhammer[n + 1, 9*n]/10!^n, {n, 0, 10}] (* Paolo Xausa, Aug 08 2024 *)

a(n) = (10n)!/(n!(10!)^n). - Christian G. Bower, Sep 15 1998
a(n) = a(n-1)*binomial(10*n-1,9). - Christian Krause, Dec 07 2023
a(n) ~ 2^(2*n+1/2) * 5^(8*n+1/2) * (n/e)^(9*n) / 567^n. - Amiram Eldar, Aug 28 2025

a(0)=1 from Andrew Howroyd, Feb 26 2018

cp's OEIS Frontend

A369288 Array read by antidiagonals: A(n,k) = the hypergraph Catalan number C_k(n), n >= 0, k >= 1.

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A370363 Number A(n,k) of partitions of [k*n] into n sets of size k having at least one set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A025037 Number of partitions of { 1, 2, ..., 5n } into sets of size 5.

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A322252 a(0) = 1 and a(n) = (5*n)!/(5!*n!^5) for n > 0.

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A370366 Number A(n,k) of partitions of [k*n] into n sets of size k having no set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A370407 Total sum over all j in [n] of the number of partitions of [j*(n-j)] into (n-j) sets of size j.

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Maple

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A025038 Number of partitions of { 1, 2, ..., 6n } into sets of size 6.

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Mathematica

Sage

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A025040 Number of partitions of { 1, 2, ..., 8n } into sets of size 8.

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A322252 a(0) = 1 and a(n) = (5n)!/(5!n!^5) for n > 0.