A025035 -id:A025035 - cp's OEIS frontend

A360098 Square array read by antidiagonals upwards: T(n,k) is the number of ways of choosing nonnegative numbers for k n-sided dice, k >= 0, n >= 1, so that summing the faces can give any integer from 0 to n^k - 1.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 7, 1, 105, 1, 1, 1, 1, 1, 1, 71, 1, 945, 1, 1, 1, 1, 1, 10, 1, 1001, 1, 10395, 1, 1, 1, 1, 1, 3, 280, 1, 18089, 1, 135135, 1, 1, 1, 1, 1, 7, 15, 15400, 1, 398959, 1

Offset: 1

William P. Orrick, Jan 25 2023

T(n,k) depends on n only through its prime signature. (See A118914.) For example, for fixed k, T(p*q^2,k) will be the same for any pair of distinct primes p and q. Hence we may define T(n,k) = R(s(n),k), where s(n) is the prime signature of n.

Also the number of Krasner factorizations of (x^(n^k)-1) / (x-1) into k polynomials each having n nonzero terms all with coefficient +1. (Krasner and Ranulac, 1937)

			There are 3 ways to assign numbers to two 4-sided dice:
 {{0, 1, 2, 3}, {0, 4, 8, 12}},
 {{0, 1, 8, 9}, {0, 2, 4,  6}},
 {{0, 1, 4, 5}, {0, 2, 8, 10}}.
The northwest corner of T(n,k) begins:
  1   1    1    1       1          1             1 ...     (s(1)  = {})
  1   1    1    1       1          1             1 ...     (s(2)  = {1})
  1   1    1    1       1          1             1 ...     (s(3)  = {1})
  1   1    3   15     105        945         10395 ...     (s(4)  = {2})
  1   1    1    1       1          1             1 ...     (s(5)  = {1})
  1   1    7   71    1001      18089        398959 ...     (s(6)  = {1,1})
  1   1    1    1       1          1             1 ...     (s(7)  = {1})
  1   1   10  280   15400    1401400     190590400 ...     (s(8)  = {3})
  1   1    3   15     105        945         10395 ...     (s(9)  = {2})
  1   1    7   71    1001      18089        398959 ...     (s(10) = {1,1})
  1   1    1    1       1          1             1 ...     (s(11) = {1})
  1   1   42  3660 614040  169200360   69444920160 ...     (s(12) = {1,2})
  1   1    1    1       1          1             1 ...     (s(13) = {1})
  1   1    7   71    1001      18089        398959 ...     (s(14) = {1,1})
  1   1    7   71    1001      18089        398959 ...     (s(15) = {1,1})
  1   1   35 5775 2627625 2546168625 4509264634875 ...     (s(16) = {4})
  ...

M. Krasner and B. Ranulac, Sur une propriété des polynomes de la division du cercle, Comptes Rendus Académie des Sciences Paris, 240:397-399, 1937.
Matthew C. Lettington and Karl Michael Schmidt, Divisor Functions and the Number of Sum Systems, arXiv:1910.02455 [math.NT], 2019.

For rows of index n = p^j, p prime, or equivalently, for rows of signature {j} we have T(p^2,k) = R({2},k) = A001147(k), T(p^3,k) = R({3},k) = A025035(k), T(p^4,k) = R({4},k) = A025036(k), and, generally, T(p^j,k) = R({j},k) = the k-th element of the j-th column of the square array A060540.
For n = p * q, p and q distinct primes, we have T(p*q,k) = R({1,1},k) = |A002119(k)|.
Column correspondences are T(n,2) = A273013(n) and T(n,3) = A131514(n).
Cf. A118914.

@cached_function
def r(i,M):
    kminus1 = len(M)
    u = tuple([1 for j in range(kminus1)])
    if i > 1 and M == u:
        return(1)
    elif M != u:
        divList = divisors(i)[:-1]
        return(sum(r(M[j],tuple(sorted(M[:j]+tuple([d])+M[j+1:]))) for d in divList for j in range(kminus1)))
def T(n,k):
    if n == 1 or k == 0:
        return(1)
    else:
        return(r(n,tuple([n for j in range(k-1)]))) / factorial(k-1)

Use M to denote a (k-1)-element multiset of positive integers. Let U denote the (k-1)-element multiset whose elements all equal 1 and let N denote the (k-1)-element multiset whose elements all equal n. For i in M, let M_{i,j} denote the result of replacing i with j in M. Then T(1,k) = T(n,0) = 1, while for n > 1 and k > 0 we have T(n,k) = r(n,N) / (k-1)! where r(i,M) is given by the recurrence
   r(i,U) = 1 for i > 1,
   r(i,M) = Sum_{m in M} Sum_{d|i,d

A370347 Number T(n,k) of partitions of [3n] into n sets of size 3 having exactly k sets {3j-2,3j-1,3j} (1<=j<=n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 9, 0, 1, 252, 27, 0, 1, 14337, 1008, 54, 0, 1, 1327104, 71685, 2520, 90, 0, 1, 182407545, 7962624, 215055, 5040, 135, 0, 1, 34906943196, 1276852815, 27869184, 501795, 8820, 189, 0, 1, 8877242235393, 279255545568, 5107411260, 74317824, 1003590, 14112, 252, 0, 1

Offset: 0

Alois P. Heinz, Feb 15 2024

			T(2,0) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
T(2,2) = 1: 123|456.
Triangle T(n,k) begins:
            1;
            0,          1;
            9,          0,        1;
          252,         27,        0,      1;
        14337,       1008,       54,      0,    1;
      1327104,      71685,     2520,     90,    0,   1;
    182407545,    7962624,   215055,   5040,  135,   0, 1;
  34906943196, 1276852815, 27869184, 501795, 8820, 189, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Row sums give A025035.
Column k=0 gives A370357.
T(n+1,n-1) gives A027468.
T(n+2,n-1) gives 252*A000292.
Cf. A023531, A055140, A370358.

b:= proc(n) option remember; `if`(n<3, [1, 0, 9][n+1],
      9*(n*(n-1)/2*b(n-1)+(n-1)^2*b(n-2)+(n-1)*(n-2)/2*b(n-3)))
    end:
T:= (n, k)-> b(n-k)*binomial(n, k):
seq(seq(T(n, k), k=0..n), n=0..10);

T(n,k) = binomial(n,k) * A370357(n-k).
Sum_{k=1..n} T(n,k) = A370358(n).
T(n,k) mod 9 = A023531(n,k).

A110103 a(n) is the number of 2-regular 4-hypergraphs on 2n labeled vertices. (In a r-hypergraph, each hyper-edge is a proper r-set; k-regular implies that each vertex is in exactly k hyperedges.)

Original entry on oeis.org

1, 0, 0, 15, 1855, 469980, 214402650, 160081596675, 182667234224475, 302414315250247200, 697372026302486234700, 2167773244010692751057625, 8842276105055583472501844625, 46275602006744820263447546152500

Offset: 0

Marni Mishna, Jul 11 2005

P-recursive.

			One of the 15 2-regular 4-hypergraphs on 6 vertices: {{1234},{4561}, {2356}}.

Marni Mishna, Maple worksheet

Cf. A025035, A110100, A110101, A025036.

Differential equation satisfied by exponential generating function sum a(n) t^(2n)/(2n)! {F(0) = 1, -144*t^3*(-2 + t^2)^2*(d^2/dt^2)F(t) - 12*(-2 + t^2)*(2*t^8-t^6 + 72 + 6*t^4-108*t^2)*(d/dt)F(t) - t^5*(-2 + t^2)*(t^2-3)*(t^4 + 4*t^2 + 36)*F(t)}.
Linear recurrence for a(n): {(15067980*n + 10550232*n^6 + 2859384*n^7 + 522720*n^8 + 128*n^11 + 2494800 + 61600*n^9 + 4224*n^10 + 52629038*n^3 + 45995730*n^4 + 26679070*n^5 + 37729494*n^2)*a(n) + (3791790*n + 109368*n^6 + 13872*n^7 + 1008*n^8 + 1247400 + 32*n^9 + 3747208*n^3 + 1767087*n^4 + 543858*n^5 + 4994577*n^2)*a(n + 1) + (28354500*n + 154560*n^6 + 11712*n^7 + 384*n^8 + 15478428*n^3 + 5309976*n^4 + 1152480*n^5 + 27874680*n^2 + 12474000)*a(n + 2) + (-623700-794025*n-48*n^6-115380*n^3-17760*n^4-1440*n^5-416757*n^2)*a(n + 3) + (599130*n + 534600 + 267282*n^2 + 59328*n^3 + 6552*n^4 + 288*n^5)*a(n + 4) + (-14166*n-26730-2484*n^2-144*n^3)*a(n + 5) + 54*a(n + 6), a(3) = 15, a(4) = 1855, a(5) = 469980, a(0) = 1, a(1) = 0, a(2) = 0}
Recurrence (of order 5): 54*(3*n - 4)*a(n) = 18*(n-1)*(2*n - 1)*(12*n^2 - 16*n + 9)*a(n-1) + 18*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(3*n + 1)*a(n-2) + 3*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(24*n^2 - 65*n + 24)*a(n-3) + 3*(n-3)*(n-2)*(n-1)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n + 1)*a(n-4) + 2*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n - 1)*a(n-5). - Vaclav Kotesovec, Mar 11 2014
a(n) ~ 2^(3*n+1) * n^(3*n) / (3^n * exp(3*n+3/2)). - Vaclav Kotesovec, Mar 11 2014

Replaced broken link, Vaclav Kotesovec, Mar 11 2014

A135458 Number of transitive reflexive binary relations R on n labeled elements where max_{x}(|{y : xRy}|)=3.

Original entry on oeis.org

0, 0, 0, 16, 148, 1805, 23700, 351239, 5919312, 112984855, 2429692570, 58481205365, 1564981962720, 46269631044377, 1502736397861062, 53336395962363115, 2059205384354896000, 86117408372404734527, 3886421055028996900050, 188615545123265662965785

Offset: 0

Alois P. Heinz, Dec 15 2007

nonn

			a(3)=16 because there are 16 relations of the given kind for 3 elements:
1R2, 2R1, 1R3, 3R1, 2R3, 3R2;
1R2, 1R3, 2R3, 3R2;
2R1, 2R3, 1R3, 3R1;
3R1, 3R2, 1R2, 2R1;
1R2, 2R1, 1R3, 2R3;
1R3, 3R1, 1R2, 3R2;
2R3, 3R2, 2R1, 3R1;
1R2, 2R3, 1R3;
1R3, 3R2, 1R2;
2R1, 1R3, 2R3;
2R3, 3R1, 2R1;
3R1, 1R2, 3R2;
3R2, 2R1, 3R1;
1R2, 1R3;
2R1, 2R3;
3R1, 3R2;
(the reflexive relationships 1R1, 2R2, 3R3 have been omitted for brevity)

A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A135429, A135312, A025035, A008277, A006882, A007318, A000248, A000142.

A025035:= proc(n) option remember; (3*n)! /n! /(6^n); end:
z:= proc(n) option remember; add(binomial(n, k+k) *doublefactorial(k+k-1) *k^(n-k-k), k=0..floor(n/2)); end:
r:= proc(n) option remember; n! * add(add(add(add(Stirling2(e, d) *a^(d+i) *(a*(a+1)/2)^(n-i-i-e-d-a) /a! /(n-i-i-e-d-a)! /i! /e! /(2^i), a=0..(n-i-i-e-d)), d=0..min(e, n-i-i-e)), e=0..(n-i-i)), i=0..floor(n/2)) end:
A135429:= proc(n) option remember; n! *add(add(A025035(i) *z(j) *r(n-3*i-j) /(3*i)! /j! /(n-3*i-j)!, j=0..(n-3*i)), i=0..floor(n/3)) end:
A000248:= proc(n) add(binomial(n, i)*(n-i)^i, i=0..n) end:
A135312:= proc(n) option remember; add(binomial(n, i+i)*doublefactorial(i+i-1)*A000248(n-i-i), i=0..floor(n/2)) end:
a:= proc(n) A135429(n)-A135312(n) end:
seq(a(i), i=0..30);

Unprotect[Power]; 0^0 = 1; A025035[n_] := A025035[n] = (3n)!/n!/6^n; z[n_] := z[n] = Sum[Binomial[n, k+k]*(k+k-1)!!*k^(n-k-k), {k, 0, Floor[n/2]}]; r[n_] := r[n] = n!*Sum[Sum[Sum[Sum[StirlingS2[e, d]*a^(d+i)*(a*(a+1)/2)^(n-i-i-e-d-a)/a!/(n-i-i-e-d-a)!/i!/e!/2^i, {a, 0, n-i-i-e-d}], {d, 0, Min[e, n-i-i-e]}], {e, 0, n-i-i}], {i, 0, Floor[n/2]}]; A135429[n_] := A135429[n] = n!*Sum[Sum[A025035[i]*z[j]*r[n-3*i-j]/(3 i)!/j!/(n-3*i-j)!, {j, 0, n-3*i}], {i, 0, Floor[n/3]}]; A135312[n_] := SeriesCoefficient[Exp[x*Exp[x]+x^2/2], {x, 0, n}]*n!; a[n_] := A135429[n]-A135312[n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz*)

a(n) = A135429(n) - A135312(n).

A185148 Number of rectangular arrangements of [1,3n] in 3 increasing sequences of size n and n monotonic sequences of size 3.

Original entry on oeis.org

1, 6, 53, 587, 7572, 109027, 1705249, 28440320, 499208817, 9134237407, 172976239886, 3371587949969, 67351686970929, 1374179898145980, 28557595591148315, 603118526483125869, 12920388129877471030, 280324904918707937001, 6151595155000424589327, 136384555249451824930126

Offset: 1

Olivier Gérard, Feb 15 2011

nonn

a(n) counts a subset of A025035(n).
a(n) counts a more general set than A005789(n).
a(n) is also the number of (3*n-1)-step walks on 3-dimensional cubic lattice from (1,0,0) to (n,n,n) with steps in {(1,0,0), (0,1,0), (0,0,1)} such that for each point (x,y,z) we have x<=y<=z or x>=y>=z. - Alois P. Heinz, Feb 29 2012

			For n = 2 the a(2) = 6 arrangements are:
+---+  +---+  +---+  +---+  +---+  +---+
|1 4|  |1 6|  |1 3|  |1 3|  |1 2|  |1 2|
|2 5|  |2 5|  |2 5|  |2 4|  |3 5|  |3 4|
|3 6|  |3 4|  |4 6|  |5 6|  |4 6|  |5 6|
+---+  +---+  +---+  +---+  +---+  +---+
Only the second of these arrangements is not counted by A005789(2).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..700 (terms 0..200 from Alois P. Heinz)

Cf. A025035, A005789.

Column k=3 of A208615. - Alois P. Heinz, Feb 29 2012

b:= proc(x, y, z) option remember;
      `if`(x=z, `if`(x=0, 1, 2*b(x-1, y, z)), `if`(x>0, b(x-1, y, z), 0)+
      `if`(y>x, b(x, y-1, z), 0)+ `if`(z>y, b(x, y, z-1), 0))
    end:
a:= n-> b(n-1, n$2):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 29 2012

b[x_, y_, z_] := b[x, y, z] = If[x == z, If[x == 0, 1, 2*b[x - 1, y, z]], If[x > 0, b[x - 1, y, z], 0] + If[y > x, b[x, y - 1, z], 0] + If[z > y, b[x, y, z - 1], 0]];
a[n_] := b[n - 1, n, n];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 12 2017, after Alois P. Heinz *)

a(n) ~ c * 27^n / n^4, where c = 0.608287207375... . - Vaclav Kotesovec, Sep 03 2014, updated Sep 07 2016

More terms and example from Alois P. Heinz, Feb 22 2011

Extended beyond a(8) by Alois P. Heinz, Feb 22 2012

A268163 Number of labeled binary-ternary rooted non-planar trees, indexed by number of leaves.

Original entry on oeis.org

0, 1, 1, 4, 25, 220, 2485, 34300, 559405, 10525900, 224449225, 5348843500, 140880765025, 4063875715900, 127418482316125, 4314607214417500, 156920190449147125, 6100643259005795500, 252476539015516440625, 11081983532721088487500, 514215436341672155715625

Offset: 0

Murray R. Bremner, Jan 27 2016

nonn

This can also be interpreted as the number of multilinear monomials of degree n in a nonassociative algebra with an (anti)commutative binary operation and a completely (skew-)symmetric ternary operation; the number of variables in the monomial corresponds to the number of leaves in the tree.

This sequence also enumerates a certain class of Feynman diagrams; see the references, links, and crossrefs below.

			For n = 4 and using the monomial interpretation, the 25 multilinear monomials of degree 4 are as follows, where [-,-] is the binary operation and (-,-,-) is the ternary operation:
[[[a,b],c],d], [[[a,b],d],c], [[[a,c],b],d], [[[a,c],d],b], [[[a,d],b],c], [[[a,d],c],b], [[[b,c],a],d], [[[b,c],d],a], [[[b,d],a],c], [[[b,d],c],a], [[[c,d],a],b], [[[c,d],b],a], [[a,b],[c,d]], [[a,c],[b,d]], [[a,d],[b,c]], [(a,b,c),d], [(a,b,d),c], [(a,c,d),b], [(b,c,d),a], ([a,b],c,d), ([a,c],b,d), ([a,d],b,c), ([b,c],a,d), ([b,d],a,c), ([c,d],a,b).

J. Bedford, On Perturbative Field Theory and Twistor String Theory, Ph.D. Thesis, 2007, Queen Mary, University of London.
B. Feng and M. Luo, An introduction to on-shell recursion relations, Review Article, Frontiers of Physics, October 2012, Volume 7, Issue 5, pp. 533-575.
K. Kampf, A new look at the nonlinear sigma model, 17th International Conference in Quantum Chromodynamics (QCD 14), Nuclear and Particle Physics Proceedings, Volumes 258-259, January-February 2015, pp. 86-89.
M. L. Mangano and S. J. Parke, Multi-parton amplitudes in gauge theories, Physics Reports, Volume 200, Issue 6, February 1991, pp. 301-367.

Alois P. Heinz, Table of n, a(n) for n = 0..400.
J. Bedford, On Perturbative Field Theory and Twistor String Theory, arXiv:0709.3478 [hep-th], 2007, (see page 23).
D. Carmi, TeV scale strings and scattering amplitudes at the LHC, arXiv:1109.5161 [hep-th], 2011-2015, (see page 23).
B. Feng, M. Luo, An introduction to on-shell recursion relations, arXiv:1111.5759 [hep-th], 2011-2012, (see page 2).
K. Kampf, A new look at the nonlinear sigma model (see pages 7 and 13).
M. L. Mangano, S. J. Parke, Multi-parton amplitudes in gauge theories, arXiv:hep-th/0509223, 2005, (see page 5).
G. Travaglini, Harmony of (super) form factors (see page 3).
A. Volovich, Yang-Mills amplitudes and twistor string theory (see bottom of page 1).
James Christopher Whitehead, The Production of Pairs of Isolated Photons at Higher Orders in QCD, Ph. D. Thesis, Durham University (UK, 2021).
Wikipedia, Feynman diagram

Cf. A001147. The number of labeled binary rooted non-planar trees.
Cf. A025035. The number of labeled ternary rooted non-planar trees.
Cf. A268172. The corresponding number of unlabelled trees.
Cf. A005411. Number of non-vanishing Feynman diagrams of order 2n for the electron or the photon propagators in quantum electrodynamics.
Cf. A005412. Number of non-vanishing Feynman diagrams of order 2n for the vacuum polarization (the proper two-point function of the photon) and for the self-energy (the proper two-point function of the electron) in quantum electrodynamics (QED).
Cf. A005413. Number of non-vanishing Feynman diagrams of order 2n+1 for the electron-electron-photon proper vertex function in quantum electrodynamics (QED).
Cf. A005414. Feynman diagrams of order 2n with vertex skeletons.
Other sequences related to Feynman diagrams: A115974, A122023, A167872, A214298, A214299.
Cf. A000311.

with(combinat):
b:= proc(n, i, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or nAlois P. Heinz, Jan 28 2016
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [0, 1$2][n+1],
       ((24*n-36)*a(n-1)+(3*n-5)*(3*n-7)*a(n-2))/11)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 28 2016

a[0]=0; a[1]=1; a[2]=1; a[n_]:=a[n]=(12(2n-3)a[n-1]+(3n-5)(3n-7)a[n-2])/11; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)

a(n) = ((24*n-36)*a(n-1)+(3*n-5)*(3*n-7)*a(n-2))/11 for n>2. - Alois P. Heinz, Jan 28 2016
Because of Koszul duality for operads, the exponential generating function is the compositional inverse of the power series x-x^2/2-x^3/6 (email of Vladimir Dotsenko to Murray R. Bremner, Jan 28 2016).
a(n) ~ sqrt(9-4*sqrt(3)) * ((12+9*sqrt(3))/11)^n * n^(n-1) / (3 * exp(n)). - Vaclav Kotesovec, Feb 24 2016

A318147 Coefficients of the Omega polynomials of order 3, triangle T(n,k) read by rows with 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, -9, 10, 0, 477, -756, 280, 0, -74601, 142362, -83160, 15400, 0, 25740261, -55429920, 40900860, -12612600, 1401400, 0, -16591655817, 38999319642, -33465991104, 13440707280, -2572970400, 190590400

Offset: 0

Peter Luschny, Aug 22 2018

The name 'Omega polynomial' is not a standard name.

			[0] [1]
[1] [0,            1]
[2] [0,           -9,          10]
[3] [0,          477,        -756,          280]
[4] [0,       -74601,      142362,       -83160,       15400]
[5] [0,     25740261,   -55429920,     40900860,   -12612600,     1401400]
[6] [0, -16591655817, 38999319642, -33465991104, 13440707280, -2572970400,190590400]

All row sums are 1, alternating row sums (taken absolute) are A002115.
T(n,1) ~ A293951(n), T(n,n) = A025035(n).
A023531 (m=1), A318146 (m=2), this seq (m=3), A318148 (m=4).

# See A318146 for the missing functions.
FL([seq(CL(OmegaPolynomial(3, n)), n=0..8)]);

(* OmegaPolynomials are defined in A318146 *)
Table[CoefficientList[OmegaPolynomial[3, n], x], {n, 0, 6}] // Flatten

# See A318146 for the function OmegaPolynomial.
[list(OmegaPolynomial(3, n)) for n in (0..6)]

Omega(m, n, z) = (m*n)!*[z^(n*m)] H(m, z)^x where H(m, z) = hypergeom([], [seq(i/m, i=1..m-1)], (z/m)^m). We consider here the case m = 3 (for other cases see the cross-references).

A327412 a(n) = multinomial(3*n+2; 2, 3, 3, ..., 3) (n times '3').

Original entry on oeis.org

1, 10, 560, 92400, 33633600, 22870848000, 26072766720000, 46174869861120000, 120054661638912000000, 438679733628584448000000, 2175851478797778862080000000, 14240947928731462652313600000000, 120136636726778618934917529600000000, 1280656547507460077846220865536000000000

Offset: 0

Peter Luschny, Sep 07 2019

nonn

Cf. A014606, A327411, A025035.

a:= n-> combinat[multinomial](3*n+2, 3$n, 2):
seq(a(n), n=0..17);  # Alois P. Heinz, Sep 07 2019

multinomial[n_, k_List] := n!/Times @@ (k!);
a[n_] := multinomial[3n+2, Join[{2}, Table[3, {n}]]];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Apr 19 2025 *)

def a(n): return multinomial([2] + [3] * n)
[a(n) for n in range(15)]

a(n) = 2^(-n-1)*3^(-n)*Gamma(3*n + 3).
a(n) = (9*(n-1)^3 + 36*(n-1)^2 + 47*n - 27)*a(n-1)/2 for n > 0.
a(n) / n! = A025035(n+1).
a(n)*(n+1) = A014606(n+1).

A361948 Array read by ascending antidiagonals. A(n, k) = Product_{j=0..k-1} binomial((j + 1)*n - 1, n - 1) if n >= 1, and A(0, k) = 1 for all k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 10, 15, 1, 1, 1, 1, 35, 280, 105, 1, 1, 1, 1, 126, 5775, 15400, 945, 1, 1, 1, 1, 462, 126126, 2627625, 1401400, 10395, 1, 1, 1, 1, 1716, 2858856, 488864376, 2546168625, 190590400, 135135, 1, 1

Offset: 0

Peter Luschny, Apr 13 2023

Row n gives the leading coefficients of the set partition polynomials of type n. The sequence of these polynomial sequences starts: A097805, A048993, A156289, A291451, A291452, ...

			Array A(n, k) starts:
  [0] 1, 1,   1,       1,           1,                 1, ...
  [1] 1, 1,   1,       1,           1,                 1, ...
  [2] 1, 1,   3,      15,         105,               945, ...  A001147
  [3] 1, 1,  10,     280,       15400,           1401400, ...  A025035
  [4] 1, 1,  35,    5775,     2627625,        2546168625, ...  A025036
  [5] 1, 1, 126,  126126,   488864376,     5194672859376, ...  A025037
  [6] 1, 1, 462, 2858856, 96197645544, 11423951396577720, ...  A025038
.
Triangle A(n-k, k) starts:
  [0] 1;
  [1] 1, 1;
  [2] 1, 1,  1;
  [3] 1, 1,  1,   1;
  [4] 1, 1,  3,   1,   1;
  [5] 1, 1, 10,  15,   1, 1;
  [6] 1, 1, 35, 280, 105, 1, 1;

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.
Antoine Chambert-Loir, Combinatorics of partitions, blog post 2024.
Alexander Karpov, Generalized knockout tournament seedings, International Journal of Computer Science in Sport, vol. 17(2), 2018.

Cf. A060540 (subarray), A370407 (antidiagonal sums, row sums).
Cf. A001147 (row 2), A025035 (row 3), A025036 (row 4), A025037 (row 5), A025038 (row 6), A025039 (row 7), A025040 (row 8), A025041 (row 9).
Cf. A088218 (column 2), A060542 (column 3), A082368 (column 4), A322252 (column 5), A057599 (main diagonal).
Cf. A097805, A048993, A156289, A291451, A291452.

A := (n, k) -> mul(binomial((j + 1)*n - 1, n - 1), j = 0..k-1):
seq(seq(A(n-k, k), k = 0..n), n = 0..9);
# Alternative, using recursion:
A := proc(n, k) local P; P := proc(n, k) option remember;
if n = 0 then return x^k*k! fi; if k = 0 then 1 else add(binomial(n*k, n*j)*
P(n,k-j)*x, j=1..k) fi end: coeff(P(n, k), x, k) / k! end:
seq(print(seq(A(n, k), k = 0..5)), n = 0..6);
# Alternative, using exponential generating function:
egf := n -> ifelse(n=0, 1, exp(x^n/n!)): ser := n -> series(egf(n), x, 8*n):
row := n -> local k; seq((n*k)!*coeff(ser(n), x, n*k), k = 0..6):
for n from 0 to 6 do [n], row(n) od;  # Peter Luschny, Aug 15 2024

A[n_, k_] := Product[Binomial[n (j + 1) - 1, n - 1], {j, 0, k - 1}]; Table[A[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 13 2023 *)

def Arow(n, size):
    if n == 0: return [1] * size
    return [prod(binomial((j + 1)*n - 1, n - 1) for j in range(k)) for k in range(size)]
for n in range(7): print(Arow(n, 7))
# Alternative, using exponential generating function:
def SetPolyLeadCoeff(m, n):
    x, z = var("x, z")
    if m == 0: return 1
    w = exp(2 * pi * I / m)
    o = sum(exp(z * w ** k) for k in range(m)) / m
    t = exp(x * (o - 1)).taylor(z, 0, m*n)
    p = factorial(m*n) * t.coefficient(z, m*n)
    return p.leading_coefficient(x)
for m in range(7):
    print([SetPolyLeadCoeff(m, k) for k in range(6)])

A(n, k) = (1/k!) * [x^k] P(n, k), where P(n, k) = k!*x^k if n = 0 and otherwise 1 if k = 0 and otherwise Sum_{j=1..k} binomial(n*k, n*j)*P(n, k-j)*x.

A(n, k) = (n*k)!*[x^(n*k)] exp(x^n/n!) for n >= 1. - Peter Luschny, Aug 15 2024

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A355144 Number T(n,k) of partitions of [n] having exactly k blocks of size at least three; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.

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A360098 Square array read by antidiagonals upwards: T(n,k) is the number of ways of choosing nonnegative numbers for k n-sided dice, k >= 0, n >= 1, so that summing the faces can give any integer from 0 to n^k - 1.

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A370347 Number T(n,k) of partitions of [3n] into n sets of size 3 having exactly k sets {3j-2,3j-1,3j} (1<=j<=n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A110103 a(n) is the number of 2-regular 4-hypergraphs on 2n labeled vertices. (In a r-hypergraph, each hyper-edge is a proper r-set; k-regular implies that each vertex is in exactly k hyperedges.)

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A135458 Number of transitive reflexive binary relations R on n labeled elements where max_{x}(|{y : xRy}|)=3.

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A185148 Number of rectangular arrangements of [1,3n] in 3 increasing sequences of size n and n monotonic sequences of size 3.

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A268163 Number of labeled binary-ternary rooted non-planar trees, indexed by number of leaves.

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