A060593 -id:A060593 - cp's OEIS frontend

A164652 Triangle read by rows: Hultman numbers: a(n,k) is the number of permutations of n elements whose cycle graph (as defined by Bafna and Pevzner) contains k cycles for n >= 0 and 1 <= k <= n+1.

1, 0, 1, 1, 0, 1, 0, 5, 0, 1, 8, 0, 15, 0, 1, 0, 84, 0, 35, 0, 1, 180, 0, 469, 0, 70, 0, 1, 0, 3044, 0, 1869, 0, 126, 0, 1, 8064, 0, 26060, 0, 5985, 0, 210, 0, 1, 0, 193248, 0, 152900, 0, 16401, 0, 330, 0, 1, 604800, 0, 2286636, 0, 696905, 0, 39963, 0, 495, 0, 1, 0, 19056960, 0, 18128396, 0, 2641925, 0, 88803, 0, 715, 0, 1

Offset: 0

Anthony Labarre, Aug 19 2009

a(n,k) is also the number of ways to express a given (n+1)-cycle as the product of an (n+1)-cycle and a permutation with k cycles (see Doignon and Labarre). a(n,n+1-2k) is the number of permutations of n elements whose block-interchange distance is k (see Christie, Doignon and Labarre).
Named after the Swedish mathematician Axel Hultman. - Amiram Eldar, Jun 11 2021
a(2*n,1) is the number of spanning trees in certain graphs with 2*n+1 vertices and n*(n+1) edges (see Ishikawa, Miezaki, and Tanaka). - Tsuyoshi Miezaki, Feb 08 2023

			Triangle begins:
  n=0:  1;
  n=1:  0, 1;
  n=2:  1, 0, 1;
  n=3:  0, 5, 0, 1;
  n=4:  8, 0, 15, 0, 1;
  n=5:  0, 84, 0, 35, 0, 1;
  n=6:  180, 0, 469, 0, 70, 0, 1;
  n=7:  0, 3044, 0, 1869, 0, 126, 0, 1;
  n=8:  8064, 0, 26060, 0, 5985, 0, 210, 0, 1;
  n=9:  0, 193248, 0, 152900, 0, 16401, 0, 330, 0, 1;
  n=10: 604800, 0, 2286636, 0, 696905, 0, 39963, 0, 495, 0, 1;
  ...
From _Jon E. Schoenfield_, May 20 2023: (Start)
As a right-aligned triangle:
                                                      1; n=0
                                                   0, 1; n=1
                                                1, 0, 1; n=2
                                           0,   5, 0, 1; n=3
                                        8, 0,  15, 0, 1; n=4
                                 0,    84, 0,  35, 0, 1; n=5
                            180, 0,   469, 0,  70, 0, 1; n=6
                      0,   3044, 0,  1869, 0, 126, 0, 1; n=7
                8064, 0,  26060, 0,  5985, 0, 210, 0, 1; n=8
          0,  193248, 0, 152900, 0, 16401, 0, 330, 0, 1; n=9
  604800, 0, 2286636, 0, 696905, 0, 39963, 0, 495, 0, 1; n=10
  ...
(End)

Axel Hultman, Toric permutations, Master's thesis, Department of Mathematics, KTH, Stockholm, Sweden, 1999.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Nikita Alexeev, Anna Pologova, and Max A. Alekseyev, Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs, Journal of Computational Biology, Vol. 24, No. 2 (2017), pp. 93-105; arXiv preprint, arXiv:1503.05285 [q-bio.GN], 2015-2017.
Nikita Alexeev and Peter Zograf, Hultman numbers, polygon gluings and matrix integrals, arXiv preprint arXiv:1111.3061 [math.PR], 2011.
Nikita Alexeev and Peter Zograf, Random matrix approach to the distribution of genomic distance, Journal of Computational Biology, Vol. 21, No. 8 (2014), pp. 622-631.
Miklos Bona and Ryan Flynn, The Average Number of Block Interchanges Needed to Sort A Permutation and a recent result of Stanley, arXiv:0811.0740 [math.CO], 2008.
Miklos Bona and Ryan Flynn, The Average Number of Block Interchanges Needed to Sort A Permutation and a recent result of Stanley, Inf. Process. Lett., Vol. 109 (2009), pp. 927-931.
David A. Christie, Sorting Permutations by Block-Interchanges, Inf. Process. Lett., Vol. 60, No. 4 (1996), pp. 165-169.
Robert Cori and Gábor Hetyei, On reduced unicellular hypermonopoles, arXiv:2403.19569 [math.CO], 2024. See at p. 4.
Jean-Paul Doignon and Anthony Labarre, On Hultman Numbers, J. Integer Seq., Vol. 10 (2007), Article 07.6.2, 13 pages.
Simona Grusea and Anthony Labarre, The distribution of cycles in breakpoint graphs of signed permutations, arXiv:1104.3353 [cs.DM], 2011-2012.
Reina Ishikawa, Tsuyoshi Miezaki, and Yuuho Tanaka, A new interpretation of Hultman numbers.

Cf. A000142 (row sums), A000217, A060593, A128174, A130534, A132393, A185259, A189507, A260695.

Cf. A185263 (rows reversed without 0's).

a164652 n k = a164652_tabl !! n !! k
a164652_row n = a164652_tabl !! n
a164652_tabl = [0] : tail (zipWith (zipWith (*)) a128174_tabl $
   zipWith (map . flip div) (tail a000217_list) (map init $ tail a130534_tabl))
-- Reinhard Zumkeller, Aug 01 2014

A164652:= (n, k)-> `if`(n-k mod 2 = 1, -Stirling1(n+2, k)/binomial(n+2, 2), 0):
for n from 0 to 7 do seq(A164652(n,k),k=1..n+1) od; # Peter Luschny, Mar 22 2015

T[n_, k_] := If[OddQ[n-k], Abs[StirlingS1[n+2, k]]/Binomial[n+2, 2], 0];
Table[T[n, k], {n, 0, 11}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)

T(n,k)= my(s=(n-k)%2); (-1)^s*s*stirling(n+2,k,1)/binomial(n+2,2);
concat(vector(12, n, vector(n, k, T(n-1,k)))) \\ Gheorghe Coserea, Jan 23 2018

def A164652(n, k):
    return stirling_number1(n+2,k)/binomial(n+2,2) if is_odd(n-k) else 0
for n in (0..7): print([A164652(n,k) for k in (1..n+1)]) # Peter Luschny, Mar 22 2015

T(n,k) = S1(n+2,k)/C(n+2,2) if n-k is odd, and 0 otherwise. Here S1(n,k) are the unsigned Stirling numbers of the first kind A132393 and C(n,k) is the binomial coefficient (see Bona and Flynn).
For n > 0: T(n,k) = A128174(n+1,k) * A130534(n+1,k-1) / A000217(n+1). - Reinhard Zumkeller, Aug 01 2014
n-th row polynomial R(n,x) = (x/2)*( P(n+1,x) + (-1)^n * P(n+1,-x) ) / binomial(n+2,2), where P(k,x) = (x + 1)*(x + 2)*...*(x + k). - Peter Bala, May 14 2023

T(0,1) set to 1 by Peter Luschny, Mar 24 2015

Edited to match values of k to the range 1 to n+1. - Max Alekseyev, Nov 20 2020

A090586 Denominator of Sum/Product of first n numbers.

Original entry on oeis.org

1, 2, 1, 12, 8, 240, 180, 1120, 8064, 725760, 604800, 79833600, 68428800, 830269440, 10897286400, 2615348736000, 2324754432000, 711374856192000, 640237370572800, 11585247657984000, 221172909834240000, 102181884343418880000, 93666727314800640000

Offset: 1

Reinhard Zumkeller, Dec 03 2003

a(n) = A000142(n)/A069268(n);
a(p-1) = 2*A000142(p-2) for odd primes.
a(n) = A060593((n - 1)/2) for odd n. - Gregory Gerard Wojnar, Jun 10 2021

Robert Israel, Table of n, a(n) for n = 1..451

Cf. A090585 (numerator), A060593.

Main diagonal of A093420.

[Denominator(n + 1) / (2*Factorial(n - 1)): n in [1..30]]; // Vincenzo Librandi, Oct 15 2018

seq(denom((n+1)/(2*(n-1)!)),n=1..25); # Robert Israel, Oct 14 2018

Table[Denominator[(n + 1) / (2 (n - 1)!)], {n, 25}] (* Vincenzo Librandi, Oct 15 2018 *)

A177267 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having genus k (see first comment for definition of genus).

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 14, 10, 0, 0, 42, 70, 8, 0, 0, 132, 420, 168, 0, 0, 0, 429, 2310, 2121, 180, 0, 0, 0, 1430, 12012, 20790, 6088, 0, 0, 0, 0, 4862, 60060, 174174, 115720, 8064, 0, 0, 0, 0, 16796, 291720, 1309308, 1624480, 386496, 0, 0, 0, 0, 0, 58786, 1385670, 9087078, 18748730, 10031736, 604800, 0, 0, 0, 0, 0, 208012, 6466460, 59306676, 188208020, 186698512, 38113920, 0

Offset: 1

Emeric Deutsch, May 27 2010

The genus g(p) of a permutation p of {1,2,...,n} is defined by g(p)=(1/2)[n+1-z(p)-z(cp')], where p' is the inverse permutation of p, c = 234...n1 = (1,2,...,n), and z(q) is the number of cycles of the permutation q.
The sum of the entries in row n is n!.
The number of nonzero entries in row n is floor((n+1)/2).
T(n,0) = A000108(n) (the Catalan numbers).
Apparently T(n,1) = A002802(n-3).
Last nonzero terms in rows with odd n appear to be A060593. [Joerg Arndt, Nov 01 2012.]

			T(3,1)=1 because 312 is the only permutation of {1,2,3} with genus 1 (we have p=312=(132), cp'=231*231=312=(132) and so g(p)=(1/2)(3+1-1-1)=1).
Triangle starts:
[ 1]  1,
[ 2]  2, 0,
[ 3]  5, 1, 0,
[ 4]  14, 10, 0, 0,
[ 5]  42, 70, 8, 0, 0,
[ 6]  132, 420, 168, 0, 0, 0,
[ 7]  429, 2310, 2121, 180, 0, 0, 0,
[ 8]  1430, 12012, 20790, 6088, 0, 0, 0, 0,
[ 9]  4862, 60060, 174174, 115720, 8064, 0, 0, 0, 0,
[10]  16796, 291720, 1309308, 1624480, 386496, 0, 0, 0, 0, 0,
[11]  58786, 1385670, 9087078, 18748730, 10031736, 604800, 0, 0, ...,
[12]  208012, 6466460, 59306676, 188208020, 186698512, 38113920, 0, ...,
[13]  742900, 29745716, 368588220, 1700309468, 2788065280, 1271140416, 68428800, 0, ...,
...

S. Dulucq and R. Simion, Combinatorial statistics on alternating permutations, J. Algebraic Combinatorics, 8, 1998, 169-191.

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A178514 (genus of derangements), A178515 (genus of involutions), A178516 (genus of up-down permutations), A178517 (genus of non-derangement permutations), A178518 (permutations of [n] having genus 0 and p(1)=k), A185209 (genus of connected permutations), A218538 (genus of permutations avoiding [x,x+1]).
Row sums give A000142.
T(2n+1,n) gives A060593.
Cf. A000108, A002802.

n := 8: with(combinat): P := permute(n): inv := proc (p) local j, q: for j to nops(p) do q[p[j]] := j end do: [seq(q[i], i = 1 .. nops(p))] end proc: nrcyc := proc (p) local nrfp, pc: nrfp := proc (p) local ct, j: ct := 0: for j to nops(p) do if p[j] = j then ct := ct+1 else end if end do: ct end proc: pc := convert(p, disjcyc): nops(pc)+nrfp(p) end proc: b := proc (p) local c: c := [seq(i+1, i = 1 .. nops(p)-1), 1]: [seq(c[p[j]], j = 1 .. nops(p))] end proc: gen := proc (p) options operator, arrow: (1/2)*nops(p)+1/2-(1/2)*nrcyc(p)-(1/2)*nrcyc(b(inv(p))) end proc: f[n] := sort(add(t^gen(P[j]), j = 1 .. factorial(n))): seq(coeff(f[n], t, j), j = 0 .. ceil((1/2)*n)-1); # yields the entries in the specified row n
# second Maple program:
b:= proc(n) option remember; `if`(n<2, 1, ((4*n-2)*
      b(n-1)+(n-2)*(n-1)^2*expand(x*b(n-2)))/(n+1))
    end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=0..n-1), n=1..12);  # Alois P. Heinz, Feb 16 2024

T[ n_, k_] := If[ n < 1 || k >= n, 0, Module[{pn = Table[i, {i, n}]}, Do[ pn[[i]] = ((4 i - 2) pn[[i - 1]] + x (i - 2) (i - 1)^2 pn[[i - 2]])/(i + 1) // Expand, {i, 3, n}]; Coefficient[pn[[n]], x, k]]]; (* Michael Somos, Sep 02 2017 *)

Let p(n, x) := g.f. of row n. Then (n+1) * p(n, x) = (4*n-2) * p(n-1, x) + x * (n-2) * (n-1)^2 * p(n-2, x). - Michael Somos, Sep 02 2017

Terms for rows 12 and 13 from Joerg Arndt, Jan 24 2011.

A321710 Triangle read by rows: T(n,k) is the number of rooted hypermaps of genus k with n darts.

Original entry on oeis.org

1, 3, 12, 1, 56, 15, 288, 165, 8, 1584, 1611, 252, 9152, 14805, 4956, 180, 54912, 131307, 77992, 9132, 339456, 1138261, 1074564, 268980, 8064, 2149888, 9713835, 13545216, 6010220, 579744, 13891584, 81968469, 160174960, 112868844, 23235300, 604800, 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880, 608583680, 5702382933, 19588944336, 28540603884, 16497874380, 2936606400, 68428800, 4107939840, 47168678571, 206254571236, 404562365316, 344901105444, 108502598960, 8099018496

Offset: 1

Gheorghe Coserea, Nov 17 2018

Row n contains floor((n+1)/2) = A008619(n-1) terms.

			Triangle starts:
n\k  [0]       [1]        [2]         [3]         [4]        [5]
[1]  1;
[2]  3;
[3]  12,       1;
[4]  56,       15;
[5]  288,      165,       8;
[6]  1584,     1611,      252;
[7]  9152,     14805,     4956,       180;
[8]  54912,    131307,    77992,      9132;
[9]  339456,   1138261,   1074564,    268980,     8064;
[10] 2149888,  9713835,   13545216,   6010220,    579744;
[11] 13891584, 81968469,  160174960,  112868844,  23235300,  604800;
[12] 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880;
[13] ...

Gheorghe Coserea, Rows n = 1..42, flattened
Alain Giorgetti and Timothy R. S. Walsh, Enumeration of hypermaps of a given genus, Ars Math. Contemp. 15 (2018) 225-266.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
P. G. Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, International Mathematics Research Notices, Volume 2015, Issue 24, 1 January 2015, 13533-13544.
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Columns k=0..9 give: A000257 (k=0), A118093 (k=1), A214817 (k=2), A214818 (k=3), A318104 (k=4), A321705 (k=5), A321706 (k=6), A321707 (k=7), A321708 (k=8), A321709 (k=9).
Row sums give A003319(n+1).
Cf. A008619, A060593.

l1[f_,n_] := Sum[(i-1)t[i]D[f,t[i-1]], {i,2,n}];
m1[f_,n_] := Sum[(i-1)t[j]t[i-j]D[f,t[i-1]] + j(i-j)t[i+1]D[f,t[j],t[i-j]], {i,2,n},{j,i-1}];
ff[1] = x^2 t[1];
ff[n_] := ff[n] = Simplify@(2x l1[ff[n-1],n] + m1[ff[n-1],n] + Sum[t[i+1]j(i-j)D[ff[k],t[j]]D[ff[n-1-k],t[i-j]], {i,2,n-1},{j,i-1},{k,n-2}]) / n;
row[n_]:=Reverse[CoefficientList[n ff[n] /. {t[_]->x}, x]][[;;;;2]][[;;Quotient[n+1,2]]];
Table[row[n], {n,14}] (* Andrei Zabolotskii, Jun 27 2025, after the PARI code *)

L1(f, N) = sum(i=2, N, (i-1)*t[i]*deriv(f, t[i-1]));
M1(f, N) = {
  sum(i=2, N, sum(j=1, i-1, (i-1)*t[j]*t[i-j]*deriv(f, t[i-1]) +
      j*(i-j)*t[i+1]*deriv(deriv(f, t[j]), t[i-j])));
};
F(N) = {
  my(u='x, v='x, f=vector(N)); t=vector(N+1, n, eval(Str("t", n)));
  f[1] = u*v*t[1];
  for (n=2, N, f[n] = (u + v)*L1(f[n-1], n) + M1(f[n-1], n) +
    sum(i=2, n-1, t[i+1]*sum(j=1, i-1,
    j*(i-j)*sum(k=1, n-2, deriv(f[k], t[j])*deriv(f[n-1-k], t[i-j]))));
    f[n] /= n);
  f;
};
seq(N) = {
  my(f=F(N), v=substvec(f, t, vector(#t, n, 'x)),
     g=vector(#v, n, Polrev(Vec(n * v[n]))));
  apply(p->Vecrev(substpol(p, 'x^2, 'x)), g);
};
concat(seq(14))

A000257(n)=T(n,0), A118093(n)=T(n,1), A214817(n)=T(n,2), A214818(n)=T(n,3), A060593(n)=T(2*n+1,n)=(2*n)!/(n+1), A003319(n+1)=Sum_{k=0..floor((n-1)/2)} T(n,k).

A268438 Triangle read by rows, T(n,k) = (-1)^k(2n)!*P[n,k](n/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 8, 6, 0, 180, 240, 90, 0, 8064, 14560, 10080, 2520, 0, 604800, 1330560, 1285200, 604800, 113400, 0, 68428800, 173638080, 209341440, 139708800, 49896000, 7484400, 0, 10897286400, 30858347520, 43770767040, 36970053120, 18918900000, 5448643200, 681080400

Offset: 0

Peter Luschny, Mar 07 2016

The P-transform is defined in the link. Compare also the Sage and Maple implementations below.

			Triangle starts:
[1],
[0, 1],
[0, 8,        6],
[0, 180,      240,       90],
[0, 8064,     14560,     10080,     2520],
[0, 604800,   1330560,   1285200,   604800,    113400],
[0, 68428800, 173638080, 209341440, 139708800, 49896000, 7484400].

Peter Luschny, The P-transform.
Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 3.

Cf. A000680, A060593, A268437, A269940, A269941.

A268438 := proc(n,k) local F,T;
  F := proc(n,k) option remember;
  `if`(n=0 and k=0, 1,`if`(n=k, (4*n-2)*F(n-1,k-1),
  F(n-1,k)*(n+k))) end;
  T := proc(n, k) option remember;
  `if`(k=0 and n=0, 1,`if`(k<=0 or k>n, 0,
  (4*n-2)*n*(n+k-1)*(T(n-1,k)+T(n-1,k-1)))) end:
T(n,k)/F(n,k) end:
for n from 0 to 6 do seq(A268438(n,k), k=0..n) od;
# Alternatively, with the function PTrans defined in A269941:
A268438_row := n -> PTrans(n, n->n/(n+1),(n,k)->(-1)^k*(2*n)!):
seq(lprint(A268438_row(n)), n=0..8);

T[n_, k_] := (2n)!/FactorialPower[n+k, n] Sum[(-1)^(m+k) Binomial[n+k, n+m] Abs[StirlingS1[n+m, m]], {m, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] (* Jean-François Alcover, Jun 15 2019 *)

A268438 = lambda n,k: (factorial(2*n)/falling_factorial(n+k,n))*sum((-1)^(m+k)* binomial(n+k,n+m)*stirling_number1(n+m,m) for m in (0..k))
for n in (0..7): print([A268438(n,m) for m in (0..n)])

# uses[PtransMatrix from A269941]
PtransMatrix(7, lambda n: n/(n+1), lambda n,k: (-1)^k*factorial(2*n))

T(n,k) = ((2*n)!/FF(n+k,n))*Sum_{m=0..k}(-1)^(m+k)*C(n+k,n+m)*Stirling1(n+m,m) where FF denotes the falling factorial function.
T(n,k) = ((2*n)!/FF(n+k,n))*A269940(n,k).
T(n,1) = (2*n)!/(n+1) = A060593(n) for n>=1.
T(n,n) = (2*n)!/2^n = A000680(n) for n>=0.

A329070 Array read by ascending antidiagonals: T(n, k) = (kn)!/(k^n(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x(x + 1)...*(x + n - 1) is the Pochhammer symbol.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 8, 6, 1, 1, 48, 180, 24, 1, 1, 384, 12960, 8064, 120, 1, 1, 3840, 1710720, 10644480, 604800, 720, 1, 1, 46080, 359251200, 35765452800, 19813248000, 68428800, 5040, 1, 1, 645120, 109930867200, 244635697152000, 2303884477440000, 70355755008000, 10897286400, 40320, 1

Offset: 0

Petros Hadjicostas, Nov 03 2019

For information about the function W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(T(n, m+2)*((m + 2)*n + 1)) (mentioned in the Formula section below), see Theorem 3.2 in Elizalde and Noy (2003) with u = 0 and m and a in the theorem equal to our m + 1. See also the documentation of array A327722.
By using the ratio test and the Stirling approximation to the gamma function, we may show that the radius of convergence of the power series for W_m(z) is infinity (for each m >= 0). Thus, the function W_m(z) (as defined by the above power series) is entire.
If we define S(m,s) = T(n-s, s+1) for m >= 0 and 0 <= s <= m, we get the triangular array that appears in the Example section below.

			Array T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows:
  1,  1,     1,        1,           1,              1,  ...
  1,  2,     6,       24,         120,            720,  ...
  1,  8,   180,     8064,      604800,       68428800, ...
  1, 48, 12960, 10644480, 19813248000, 70355755008000, ...
  ...
Triangular array S(m,s) = T(m-s, s+1) (with rows m >= 0 and columns s >= 0):
  1;
  1,     1;
  1,     2,         1;
  1,     8,         6,           1;
  1,    48,       180,          24,           1;
  1,   384,     12960,        8064,         120,        1;
  1,  3840,   1710720,    10644480,      604800,      720,    1;
  1, 46080, 359251200, 35765452800, 19813248000, 68428800, 5040, 1;
  ...

Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125; see Theorem 3.2 (p. 116).
Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Wikipedia, Falling and rising factorials.

Rows include A000012 (n = 0), A000142 (n = 1), A060593 (n = 2).
Columns include A000012 (k = 1), A000165 (k = 2), A176730 (k = 3).
Ratios T(n+1,k)/(k!*T(n,k)) include A000012 (k = 1), A000027 (k = 2), A000326 (k = 3), A100157 (k = 4), A234043 (k = 5).
Cf. A111004, A117226, A202213, A327722.

A := (n, k) -> `if`(k=0, 1, (GAMMA(1/k)*GAMMA(k*n+1))/(GAMMA(n+1/k)*k^n)):
seq(seq(A(n-k-1, k), k=1..n-1), n=0..10); # Peter Luschny, Nov 04 2019

T(0,k) = 1, T(1,k) = k!, and T(2,k) = (2*k)!/(k + 1) for k >= 1.
T(n,1) = 1, T(n,2) = (2*n)!!, and T(n,3) is related to the Airy functions (see the documentation of A176730).
T(n+1,k) = (k-1)! * binomial(k*(n+1), k-1) * T(n,k) for n >= 0 and k >= 1.
T(n+1,k)/(k! * T(n,k)) = Cat(n+1, k), where Cat(d, k) = binomial(k*d, k)/(k * (d - 1) + 1) is a Fuss-Catalan number; see Theorem 1.2 in Schuetz and Whieldon (2014).
If F(k,z) = Sum_{n >= 0} z^(k*n)/T(n,k), then F(k,z) satisfies the o.d.e. F^(k-1)(k,z) - z*F(k,z) = 0.
If W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(T(n, m+2)*((m + 2)*n + 1)), then 1/W_m(z) is the e.g.f. of row m of A327722(m,n), which counts permutations of [n] that avoid the consecutive pattern 12...(m+1)(m+3)(m+2) (or equivalently, the consecutive pattern (m+3)(m+2)...(3)(1)(2)).
The function W_m(z) satisfies the o.d.e. W_m^(m+2)(z) + z*W_m'(z) = 0 with W_m(0) = 1, W_m'(0) = -1, and W_m^(s)(0) = 0 for s = 2..(m + 1).

A185263 Triangle T(n,k) read by rows: coefficients (in compressed forms) in order of decreasing exponents of polynomials p_n(t) related to Hultman numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 15, 8, 1, 35, 84, 1, 70, 469, 180, 1, 126, 1869, 3044, 1, 210, 5985, 26060, 8064, 1, 330, 16401, 152900, 193248, 1, 495, 39963, 696905, 2286636, 604800, 1, 715, 88803, 2641925, 18128396, 19056960, 1, 1001, 183183, 8691683, 109425316, 292271616, 68428800, 1, 1365, 355355, 25537655, 539651112, 2961802480, 2699672832

Offset: 0

N. J. A. Sloane, Jan 21 2012

Row n contains floor(n/2) + 1 terms.

			Triangle begins:
  n\k| 0    1      2       3         4         5        6
-----+---------------------------------------------------
   0 | 1
   1 | 1
   2 | 1    1
   3 | 1    5
   4 | 1   15      8
   5 | 1   35     84
   6 | 1   70    469     180
   7 | 1  126   1869    3044
   8 | 1  210   5985   26060      8064
   9 | 1  330  16401  152900    193248
  10 | 1  495  39963  696905   2286636    604800
  11 | 1  715  88803 2641925  18128396  19056960
  12 | 1 1001 183183 8691683 109425316 292271616 68428800
  ...
Polynomials p_n(t):
  p_0 = t;
  p_1 = t^2;
  p_2 = t^3 +     t;
  p_3 = t^4 +   5*t^2;
  p_4 = t^5 +  15*t^3 +    8*t;
  p_5 = t^6 +  35*t^4 +   84*t^2;
  p_6 = t^7 +  70*t^5 +  469*t^3 +  180*t;
  p_7 = t^8 + 126*t^6 + 1869*t^4 + 3044*t^2;
  ...
A(x;t) = t + t^2*x/1! + (t^3 + t)*x^2/2! + (t^4 + 5*t^2)*x^3/3! + ...

Gheorghe Coserea, Rows n = 0..202, flattened
Nikita Alexeev and Peter Zograf, Hultman numbers, polygon gluings and matrix integrals, arXiv preprint arXiv:1111.3061 [math.PR], 2011.

Cf. A038720, A048994, A060593, A185259.

For uncompressed form of polynomial coefficients, in order of increasing powers, see A164652.

T[n_, k_] := Abs[StirlingS1[n+2, n-2k+1]]/Binomial[n+2, 2];
Table[T[n, k], {n, 0, 13}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, Aug 12 2018 *)

seq(N) = {
  my(p=vector(N), t='t, v); p[1] = t^2; p[2] = t^3 + t;
  for (n=3, N,
    p[n] = ((2*n+1)*t*p[n-1] + (n-1)*(n^2-t^2)*p[n-2])/(n+2));
  v = vector(#p, n, vector(1+n\2, k, polcoeff(p[n], n+1-2*(k-1))));
  concat([[1]], v);
};
concat(seq(13))

N=14; x='x+O('x^(N+1));
concat(apply(p->select(a->a!=0, Vec(p)), Vec(serlaplace(((1-x)^(-t) - (1+x)^t)/x^2))))

T(n,k) = -stirling(n+2, n+1-2*k, 1)/binomial(n+2,2);
concat(1, concat(vector(13, n, vector(1+n\2, k, T(n, k-1)))))
\\ Gheorghe Coserea, Jan 29 2018

From Gheorghe Coserea, Jan 29 2018: (Start)
p(n) = Sum_{k=0..floor(n/2)} T(n,k)*t^(n+1-2*k) satisfies (n+2)*p(n) = (2*n+1)*t*p(n-1) + (n-1)*(n^2-t^2)*p(n-2), n >= 2. (th. 3, (iii))
E.g.f. A(x;t) = Sum_{n>=0} p(n)*x^n/n! = ((1-x)^(-t) - (1+x)^t)/x^2. (th. 3, (i))
T(n,k) = -Stirling1(n+2, n+1-2*k)/binomial(n+2,2), where Stirling1(n,k) is defined by A048994.
A000142(n) = p(n)(1), A052572(n) = p(n)(2) for n > 0, A060593(n) = T(2*n, n) for n > 0. (End)
n-th row polynomial R(n,x) satisfies x*R(n,x^2) = (1/2)*( P(n+1,x) - P(n+1,-x) )/ binomial(n+2,2), where P(k,x) = (1 + x)*(1 + 2*x) * ... *(1 + k*x). - Peter Bala, May 14 2023

More terms from Gheorghe Coserea, Jan 29 2018

A064335 a(n) = 6(2n)!/(n+2).

Original entry on oeis.org

3, 4, 36, 864, 40320, 3110400, 359251200, 58118860800, 12553673932800, 3492203839488000, 1216451004088320000, 518769566666588160000, 265906457885674045440000, 161316584450642254233600000

Offset: 0

Karol A. Penson, Sep 13 2001

nonn

All terms, except a(0) and a(1), are integer multiples of 6.

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A060593.

List([0..20], n-> 6*Factorial(2*n)/(n+2)); # G. C. Greubel, May 03 2019

[6*Factorial(2*n)/(n+2): n in [0..20]]; // G. C. Greubel, May 03 2019

Table[6*(2*n)!/(n+2), {n,0,20}] (* G. C. Greubel, May 03 2019 *)

{ s=6; for (n=0, 100, if (n, s*=2*n*(2*n - 1)); a=s/(n + 2); write("b064335.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 12 2009

a(n) = 6*(2*n)!/(n+2); \\ Michel Marcus, Jun 24 2018

[6*factorial(2*n)/(n+2) for n in (0..20)] # G. C. Greubel, May 03 2019

a(n) = Integral_{x=0..oo} (x^n*(exp(-sqrt(x)) * (-1+sqrt(x)+2/sqrt(x)) + x*Ei(-sqrt(x))) ), n=0, 1..., where Ei(y) is the exponential integral. Representation as the n-th moment of a positive function on a positive half-axis, in Maple notation. This representation is unique.

A322544 a(n) is the reciprocal of the coefficient of x^n in the power series defined by ((1+2x)exp(x) + 3exp(-x) - 4)/ (4x^2).

Original entry on oeis.org

1, 6, 8, 60, 180, 1680, 8064, 90720, 604800, 7983360, 68428800, 1037836800, 10897286400, 186810624000, 2324754432000, 44460928512000, 640237370572800, 13516122267648000, 221172909834240000, 5109094217170944000, 93666727314800640000, 2350183339898634240000, 47726800133326110720000

Offset: 0

Pierre-Alain Sallard, Dec 14 2018

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..445

Cf. A060593 (even bisection, shifted), A028242 (denominator minus 1), A030451 (denominator, shifted), A107991 (Expansion of a similar function), A073743.

List([0..25],n->(4*Factorial(n+2))/(2*n+5+3*(-1)^n)); # Muniru A Asiru, Dec 20 2018

a:=n->factorial(n+2)/(3*floor(n/2)-n+2): seq(a(n),n=0..25); # Muniru A Asiru, Dec 20 2018

Table[4*Factorial[n + 2]/(2*n + 5 + 3*(-1)^n), {n, 0, 25}]
(* or *)
Function[x, 1/x] /@
CoefficientList[Series[(Exp[x]/4 + 3/4*Exp[-x] + x/2*Exp[x] - 1)/x^2, {x, 0, 20}], x]

a(n)={(4*(n+2)!)/(5 + 3*(-1)^n + 2*n)} \\ Andrew Howroyd, Dec 14 2018

my(x='x + O('x^30)); Vec(apply(x->1/x, ((1+2*x)*exp(x) + 3*exp(-x) - 4)/ (4*x^2))) \\ Michel Marcus, Dec 19 2018

a(n) = (n+2)!/(3*floor(n/2)-n+2).
a(n) = (4*(n+2)!)/(2n+5+3*(-1)^n).
a(n) = 4/([x^n]((exp(x)*(1+2x)+3*exp(-x)-4)/x^2)).
a(n) = (n+2)!/(A028242(n)+1).
a(n) = (n+2)!/A030451(n+1).
a(n) ~ sqrt(Pi/2)/72*exp(-n)*n^(n-1/2)*(1705 - 264*n + 288*n^2). - Stefano Spezia, Aug 11 2025
Sum_{n>=0} 1/a(n) = 3*cosh(1)/2 - 1. - Amiram Eldar, Aug 15 2025

A371665 T(n,k) is the number of reduced unicellular hypermonopoles on n points with k hyperedges, where T(n,k), 1 <= k <= floor(n/2), is an array read by rows.

Original entry on oeis.org

1, 0, 1, 8, 0, 0, 36, 0, 180, 0, 49, 0, 1604, 0, 21, 8064, 0, 5144, 0, 0, 112608, 0, 7680, 0, 604800, 0, 604428, 0, 5445, 0, 11799360, 0, 1669052, 0, 1485, 68428800, 0, 91705536, 0, 2610608, 0, 0, 1741669632, 0, 384036016, 0, 2342340, 0, 10897286400, 0, 18071744976, 0, 972895560, 0, 1126125

Offset: 3

Gabor Hetyei, Apr 02 2024

T(n,k) is zero unless k <= n/2. (proven to be correct)

			The table begins:
         1;
         0,          1;
         8,          0;
         0,         36,        0;
       180,          0,       49;
         0,       1604,        0,        21;
      8064,          0,     5144,         0;
         0,     112608,        0,      7680,       0;
    604800,          0,   604428,         0,    5445;
         0,   11799360,        0,   1669052,       0,    1485;
  68428800,          0, 91705536,         0, 2610608,       0;
         0, 1741669632,        0, 384036016,       0, 2342340, 0;

Robert Cori and Gábor Hetyei, On reduced unicellular hypermonopoles, arXiv:2403.19569 [math.CO], 2024.

Cf. A060593, A164652.

proc(n, k)
    local i;
    coeff(expand(add(combinat:-binomial(n, i)*(-x)^i*(pochhammer(x, n - i + 1) - pochhammer(x - n + i, n - i + 1))/((n - i)*(n - i + 1)), i = 0 .. n - 1)), x, k);
end proc

T(n,k) = Sum_{i=0..k-1} (-1)^i binomial(n,i)*a(n-1-i,k-i) where the a(n,k) are the Hultman numbers from A164652.

T(2*m+1,1) = (2*m)! / (m+1) = A060593(m) for m >= 1.

cp's OEIS Frontend

A164652 Triangle read by rows: Hultman numbers: a(n,k) is the number of permutations of n elements whose cycle graph (as defined by Bafna and Pevzner) contains k cycles for n >= 0 and 1 <= k <= n+1.

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A090586 Denominator of Sum/Product of first n numbers.

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A177267 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having genus k (see first comment for definition of genus).

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A321710 Triangle read by rows: T(n,k) is the number of rooted hypermaps of genus k with n darts.

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A268438 Triangle read by rows, T(n,k) = (-1)^k*(2*n)!*P[n,k](n/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

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A329070 Array read by ascending antidiagonals: T(n, k) = (k*n)!/(k^n*(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x*(x + 1)*...*(x + n - 1) is the Pochhammer symbol.

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A268438 Triangle read by rows, T(n,k) = (-1)^k(2n)!*P[n,k](n/(n+1)) where P is the P-transform, for n>=0 and 0<=k<=n.

A329070 Array read by ascending antidiagonals: T(n, k) = (kn)!/(k^n(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x(x + 1)...*(x + n - 1) is the Pochhammer symbol.

A064335 a(n) = 6(2n)!/(n+2).

A322544 a(n) is the reciprocal of the coefficient of x^n in the power series defined by ((1+2x)exp(x) + 3exp(-x) - 4)/ (4x^2).