A078740 -id:A078740 - cp's OEIS frontend

A090452 Scaled array A078740 ((3,2)-Stirling2).

1, 1, 3, 2, 1, 7, 16, 15, 5, 1, 12, 51, 105, 114, 63, 14, 1, 18, 118, 396, 771, 910, 644, 252, 42, 1, 25, 230, 1110, 3235, 6083, 7580, 6240, 3270, 990, 132, 1, 33, 402, 2600, 10365, 27483, 50464, 65331, 59625, 37620, 15642, 3861, 429, 1, 42, 651, 5390, 27825, 97188

Offset: 1

Wolfdieter Lang, Dec 23 2003

This scaled Stirling2 array will be called s2_{3,2}(n,m).
The sequence of row lengths is [1,3,5,7,...]=A005408(n-1).
The generating function for the sequence from column no. m is G(m,x)=(x^ceiling(m/2))*P(m,x)/(1-x)^(2*m-3) with the row polynomials of array A091029(m,k).
The generating functions of the column sequences obey the hypergeometric differential-difference eq.:x*(1-x)*G''(m,x) + 2*(1-m*x)*G'(m,x) - m*(m-1)*G(m,x) = 2*m*x*G'(m-1,x) + 2*m*(m-1)*G(m-1,x) + m*(m-1)*G(m-2,x), m>=3; with G(2,x)=x/(1-x) and G(1,x)=0. The primes denote differentiation w.r.t. x.

			Triangle begins:
  [1];
  [1,3,2];
  [1,7,16,15,5];
  [1,12,51,105,114,63,14];
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (rows 1 <= n <= 100, flattened.)
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Wolfdieter Lang, First 8 rows.

a(n, 2*n)=A000108(n) (Catalan), n>=1, a(n, 2*n-1)=3*A002054(n-1), n>=2, a(n, 2*n-2)=A091031(n), n>=2.
The column sequences (without leading zeros) are: A000012 (powers of 1), A055998, A090453-4, A091026-7, etc.
Cf. A090442 (row sums). The alternating row sums are 0 except for row n=1 which gives 1.

Table[(-1)^m*m!*HypergeometricPFQ[{2 - m, n + 1, n + 2}, {2, 3}, 1]/(2 (m - 2)!), {n, 8}, {m, 2, 2 n}] // Flatten (* Michael De Vlieger, Nov 21 2019, after Jean-François Alcover at A078740. *)

a(n, m) = (m!/((n+1)!*n!))*A078740(n, m), n>=1, 2<= m <=2*n.

Recursion: a(n, m) = ((n+m-1)*(n+m-2)*a(n-1, m)+2*(n+m-2)*m*a(n-1, m-1)+m*(m-1)*a(n-1, m-2))/((n+1)*n), n>=2, 2<=m<=2*n, a(1, 2)=1, a(n, 0) := 0, a(n, 1) := 0 (from the recursion of array A078740).

A090437 Alternating row sums of array A078740 ((3,2)-Stirling2).

Original entry on oeis.org

1, 1, -17, -299, 1921, 451621, 23016631, -138672407, -208026131039, -31455532865879, -2214855733827329, 358045039222582141, 197933737480453452193, 51028310360637930765901, 6769736050165112232649351, -1644881699523140953828119599

Offset: 1

Wolfdieter Lang, Dec 23 2003

Cf. A078740.

a(n) = Sum_{k=2..2*n} A078740(n, k)*(-1)^k, n>=1.

A091549 Second column (k=3) sequence of array A078740 ((3,2)-Stirling2) divided by 6.

Original entry on oeis.org

1, 28, 960, 43200, 2520000, 186278400, 17069875200, 1902071808000, 253487646720000, 39833773056000000, 7291173820170240000, 1538106259064094720000, 370502654756909875200000, 101080724272141565952000000, 31008222182732149555200000000, 10627137906465962295558144000000

Offset: 2

Wolfdieter Lang, Feb 13 2004

Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.

Cf. A078740.

A091549 := proc(n)
    n!*(n+1)!*(-3 + (n+2)*(n+1)/2)/(3!)^2 ;
end proc:
seq(A091549(n),n=2..30) ; # R. J. Mathar, Jul 27 2022

a[n_] := n!*(n+1)!*((n+2)*(n+1)/2 - 3) / 36; Array[a, 16, 2] (* Amiram Eldar, Sep 01 2025 *)

a(n) = n!*(n+1)!*(-3 + (n+2)*(n+1)/2)/(3!)^2, n>=2.
E.g.f.: (hypergeom([2, 3], [], x) - 3*hypergeom([1, 2], [], x) + 2)/(3!)^2.
a(n) = Product_{j=0..n-1} (j+2) * (-3 * Product_{j=0..n-1} (j+1) + Product_{j=0..n-1} (j+3))/(3!)^2, n>=2. From eq.12 of the Blasiak et al. reference with r=3, s=2, k=3.
D-finite with recurrence a(n) + (-n^2-7*n-24)*a(n-1) + 12*(n^2+4*n+6)*a(n-2) - 36*n*(n+1)*a(n-3) = 0. - R. J. Mathar, Jul 27 2022
a(n) ~ Pi * n^(2*n+4) / (36 * exp(2*n)). - Amiram Eldar, Sep 01 2025

A091741 Coefficients of certain polynomials related to array A078740 ((3,2)-Stirling2).

Original entry on oeis.org

1, 4, 1, -36, 8, 9, 1, -288, 18, 83, 18, 1, 7200, -2352, -2366, 165, 205, 27, 1, 86400, -18000, -31936, -926, 2735, 565, 41, 1, -4233600, 1647360, 1541304, -286084, -187614, -1491, 7056, 1014, 54, 1, -67737600, 19968480, 27275064, -2562556, -3442594, -254583, 115605, 24906

Offset: 2

Wolfdieter Lang, Feb 13 2004

A078740(n,k)=(((-1)^k)/k!)*sum(((-1)^j)*binomial(k,j)*risefac(j-1,n)*risefac(j,n),j=2..k) with risefac(x,n) := Pochhammer(x,n).

The sequence of row lengths of this array is [1,2,4,5,7,8,10,11,...] = A001651(k-2) = floor((3*k-4)/2) for k>=2.

Wolfdieter Lang, First 8 rows.

P(k, n) := (-1)^k*(k-1)!*(k-2)!*(Sum_{j=2..k} (-1)^j*binomial(k, j)*risefac(j-1, n)*risefac(j, n))/(n!^2*(n+1)*Product_{p=1..ceiling(k/2)-1} (n-p)) is a polynomial in n of degree A032766(k-2), k >= 2. risefac(x, n) := Pochhammer(x, n).

a(k, m) = [n^m]P(k, n) with the above defined polynomials in n defined for k >= 2.

A078739 Triangle of generalized Stirling numbers S_{2,2}(n,k) read by rows (n>=1, 2<=k<=2n).

Original entry on oeis.org

1, 2, 4, 1, 4, 32, 38, 12, 1, 8, 208, 652, 576, 188, 24, 1, 16, 1280, 9080, 16944, 12052, 3840, 580, 40, 1, 32, 7744, 116656, 412800, 540080, 322848, 98292, 16000, 1390, 60, 1, 64, 46592, 1446368, 9196992, 20447056, 20453376, 10564304, 3047520, 511392, 50400

Offset: 1

N. J. A. Sloane, Dec 21 2002

A generalization of the Stirling2 numbers S_{1,1} from A008277.
The g.f. for column k=2*K is (x^K)*pe(K,x)*d(k,x) and for k=2*K+1 it is (x^K)*po(K,x)*2*(K+1)*K*d(k,x), K>= 1, with d(k,x) := 1/product(1-p*(p-1)*x,p=2..k) and the row polynomials pe(n,x) := sum(A089275(n,m)*x^m,m=0..n-1) and po(n,x) := sum(A089276(n,m)*x^m,m=0..n-1). - Wolfdieter Lang, Nov 07 2003
The formula for the k-th column sequence is given in A089511.
Codara et al., show that T(n,k) gives the number of k-colorings of the graph nK_2 (the disjoint union of n copies of the complete graph K_2). An example is given below. - Peter Bala, Aug 15 2013

			From _Peter Bala_, Aug 15 2013: (Start)
The table begins
n\k | 2    3    4    5    6   7   8
= = = = = = = = = = = = = = = = = =
  1 | 1
  2 | 2    4    1
  3 | 4   32   38   12    1
  4 | 8  208  652  576  188  24   1
...
Graph coloring interpretation of T(2,3) = 4: The graph 2K_2 is 2 copies of K_2, the complete graph on 2 vertices:
o---o  o---o
a   b  c   d
The four 3-colorings of 2K_2 are ac|b|d, ad|b|c, bc|a|d and bd|a|c. (End)

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
Steve Butler, Fan Chung, Jay Cummings, R. L. Graham, Juggling card sequences, arXiv:1504.01426 [math.CO], 2015.
Leonard Carlitz, On Arrays of Numbers, Am. J. Math., 54,4 (1932) 739-752. [Eqs. (3) and (4) with lambda = 0, mu = 2, a_{n,k-1} = a(n, k).- _Wolfdieter Lang_, Jan 30 2020 ]
P. Codara, O. M. D’Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers arXiv:1308.1700v1 [cs.DM]
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
S.-M. Ma, T. Mansour, M. Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.

Row sums give A020556. Triangle S_{1, 1} = A008277, S_{2, 1} = A008297 (ignoring signs), S_{3, 1} = A035342, S_{3, 2} = A078740, S_{3, 3} = A078741. A090214 (S_{4,4}).
The column sequences are A000079(n-1)(powers of 2), 4*A016129(n-2), A089271, 12*A089272, A089273, etc.
Main diagonal is A217900.
Cf. A071951 (Legendre-Stirling triangle).
Cf. A122193, A055203.

# Note that the function implements the full triangle because it can be
# much better reused and referenced in this form.
A078739 := proc(n,k) local r;
add((-1)^(n-r)*binomial(n,r)*combinat[stirling2](n+r,k),r=0..n) end:
# Displays the truncated triangle from the definition:
seq(print(seq(A078739(n,k),k=2..2*n)),n=1..6); # Peter Luschny, Mar 25 2011

t[n_, k_] := Sum[(-1)^(n-r)*Binomial[n, r]*StirlingS2[n+r, k], {r, 0, n}]; Table[t[n, k], {n, 1, 7}, {k, 2, 2*n}] // Flatten (* Jean-François Alcover, Apr 11 2013, after Peter Luschny *)

a(n, k) = sum(binomial(k-2+p, p)*A008279(2, p)*a(n-1, k-2+p), p=0..2) if 2 <= k <= 2*n for n>=1, a(1, 2)=1; else 0. Here A008279(2, p) gives the third row (k=2) of the augmented falling factorial triangle: [1, 2, 2] for p=0, 1, 2. From eq.(21) with r=2 of the Blasiak et al. paper.
a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*A008279(p, 2)^n, p=2..k) for 2 <= k <= 2*n, n>=1. From eq.(19) with r=2 of the Blasiak et al. paper.
a(n, k) = sum(A071951(n, j)*A089503(j, 2*j-k+1), j=ceiling(k/2)..min(n, k-1)), 1<=n, 2<=k<=2n; relation to Legendre-Stirling triangle. Wolfdieter Lang, Dec 01 2003
a(n, k) = A122193(n,k)*2^n/k! - Peter Luschny, Mar 25 2011
E^n = sum_{k=2}^(2n) a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^2d^2/dx^2.
The row polynomials R(n,x) are given by the Dobinski-type formula R(n,x) = exp(-x)*sum {k = 0..inf} (k*(k-1))^n*x^k/k!. - Peter Bala, Aug 15 2013

More terms from Wolfdieter Lang, Nov 07 2003

A078741 Triangle of generalized Stirling numbers S_{3,3}(n,k) read by rows (n>=1, 3<=k<=3n).

Original entry on oeis.org

1, 6, 18, 9, 1, 36, 540, 1242, 882, 243, 27, 1, 216, 13608, 94284, 186876, 149580, 56808, 11025, 1107, 54, 1, 1296, 330480, 6148872, 28245672, 49658508, 41392620, 18428400, 4691412, 706833, 63375, 3285, 90, 1, 7776, 7954848, 380841264, 3762380016, 13062960720

Offset: 1

N. J. A. Sloane, Dec 21 2002

The sequence of row lengths for this array is [1,4,7,10,..]= A016777(n-1), n>=1.
The g.f. for the k-th column, (with leading zeros and k>=3) is G(k,x)= x^ceiling(k/3)*P(k,x)/product(1-fallfac(p,3)*x,p=3..k), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := sum(A089517(k,m)*x^m,m=0..kmax(k)), k>=3, with kmax(k) := A004523(k-3)= floor(2*(k-3)/3)= [0,0,1,2,2,3,4,4,5,...]. For the recurrence of the G(k,x) see A089517. Wolfdieter Lang, Dec 01 2003
For the computation of the k-th column sequence see A090219.
Codara et al., show that T(n,k) gives the number of k-colorings of the graph nK_3 (the disjoint union of n copies of the complete graph K_3). An example is given below. - Peter Bala, Aug 15 2013

			From _Peter Bala_, Aug 15 2013: (Start)
The table begins
n\k |   3     4     5      6      7     8     9   10  11  12
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
  1 |   1
  2 |   6    18     9      1
  3 |  36   540  1242    882    243    27     1
  4 | 216 13608 94284 186876 149580 56808 11025 1107  54   1
...
Graph coloring interpretation of T(2,3) = 6:
The graph 2K_3 is 2 copies of K_3, the complete graph on 3 vertices:
    o b      o e
   / \      / \
  o---o    o---o
  a   c    d   f
The six 3-colorings of 2K_3 are ad|be|cf, ad|bf|ce, ae|bd|cf, ae|bf|cd, af|bd|ce, and af|be|cd. (End)

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027,2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Codara, O. M. D’Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers arXiv:1308.1700v1 [cs.DM]
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
W. Lang, First 6 rows.

Row sums give A069223. Cf. A078739.
The column sequences (without leading zeros) are A000400 (powers of 6), 18*A089507, 9*A089518, A089519, etc.
A089504, A069223 (row sums), A090212 (alternating row sums).
A078740, A090214.

a[n_, k_] := (-1)^k*Sum[(-1)^p*((p-2)*(p-1)*p)^n*Binomial[k, p], {p, 3, k}]/k!; Table[a[n, k], {n, 1, 6}, {k, 3, 3*n}] // Flatten (* Jean-François Alcover, Dec 04 2013 *)

a(n, k) = (((-1)^k)/k!)*Sum_{p = 3..k} (-1)^p* binomial(k, p)*fallfac(p, 3)^n, with fallfac(p, 3) := A008279(p, 3) = p*(p-1)*(p-2); 3 <= k <= 3*n, n >= 1, else 0. From eq.(19) with r = 3 of the Blasiak et al. reference.
E^n = Sum_{k = 3..3*n} a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^3d^3/dx^3.
The row polynomials R(n,x) are given by the Dobinski-type formula R(n,x) = exp(-x)*Sum_{k >= 0} (k*(k-1)*(k-2))^n*x^k/k!. - Peter Bala, Aug 15 2013

A090438 Generalized Stirling2 array (4,2).

Original entry on oeis.org

1, 12, 8, 1, 360, 480, 180, 24, 1, 20160, 40320, 25200, 6720, 840, 48, 1, 1814400, 4838400, 4233600, 1693440, 352800, 40320, 2520, 80, 1, 239500800, 798336000, 898128000, 479001600, 139708800, 23950080, 2494800, 158400, 5940, 120, 1

Offset: 1

Wolfdieter Lang, Dec 23 2003

The row length sequences for this array is [1,3,5,7,9,11,...] = A005408(n-1), n>=1.

The scaled array a(n,k)/((2*n)!/k!) = A034870(n-1,k-2), n>=1, 2<=k<=2*n (Pascal triangle, even numbered rows only).

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
Wolfdieter Lang, First 6 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A078740 (3, 2)-Stirling2.
Cf. A072678 (row sums), A090439 (alternating row sums).
Cf. A062139.

with(PolynomialTools):
p := n -> (2*n+2)!*hypergeom([-2*n],[3], -x)/2:
seq(CoefficientList(simplify(p(n)),x), n=0..5); # Peter Luschny, Apr 08 2015

a[n_, k_] := (-1)^k/k!*Sum[(-1)^p*Binomial[k, p]*Product[FactorialPower[p + 2*(j-1), 2], {j, 1, n}], {p, 2, k}]; Table[a[n, k], {n, 1, 8}, {k, 2, 2 n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

Recursion: a(n, k) = sum(binomial(2, p)*fallfac(2*(n-1)-p+k, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 2)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=4, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).
a(n, k) = (((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+2*(j-1), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=4, s=2.
a(n, k) = ((2*n)!/k!)*binomial(2*(n-1), k-2), n>=1, 2<=k<=2*n.
E.g.f. column k>=2 (with leading zeros): (((-1)^k)/k!)*(sum(((-1)^p)*binomial(k, p)*hypergeom([(p-1)/2, p/2], [], 4*x), p=2..k)-(k-1)).
Coefficient triangle of the polynomials (2*n+2)!*hypergeom([-2*n],[3],-x)/2. - Peter Luschny, Apr 08 2015
Coefficient triangle of Laguerre polynomials (2*n)!*L(2*n,2,-x). - Peter Luschny, Apr 08 2015

A091534 Generalized Stirling2 array (5,2).

Original entry on oeis.org

1, 20, 10, 1, 1120, 1040, 290, 30, 1, 123200, 161920, 71320, 14040, 1340, 60, 1, 22422400, 37452800, 22097600, 6263040, 958720, 82800, 4000, 100, 1, 6098892800, 12222918400, 8928102400, 3257116800, 675281600, 84782880, 6625920, 322000

Offset: 1

Wolfdieter Lang, Jan 23 2004

The row length sequences for this array is [1,3,5,7,9,11,...]=A005408(n-1), n>=1.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
W. Lang, First 6 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A078740 (3, 2)-Stirling2, A090438 (4, 2)-Stirling2.

Cf. A072019 (row sums), A091537 (alternating row sums).

a[n_, k_] := (-1)^k/k!*Sum[(-1)^p*Binomial[k, p]*Product[FactorialPower[p + 3*(j - 1), 2], {j, 1, n}], {p, 2, k}]; Table[a[n, k], {n, 1, 8}, {k, 2, 2 n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

a(n, k)=(((-1)^k)/k!)*sum(((-1)^p)*binomial(k, p)*product(fallfac(p+3*(j-1), 2), j=1..n), p=2..k), n>=1, 2<=k<=2*n, else 0. From eq. (12) of the Blasiak et al. reference with r=5, s=2.

Recursion: a(n, k)=sum(binomial(2, p)*fallfac(3*(n-1)+k-p, 2-p)*a(n-1, k-p), p=0..2), n>=2, 2<=k<=2*n, a(1, 2)=1, else 0. Rewritten from eq.(19) of the Schork reference with r=5, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle).

A090214 Generalized Stirling2 array S_{4,4}(n,k).

Original entry on oeis.org

1, 24, 96, 72, 16, 1, 576, 13824, 50688, 59904, 30024, 7200, 856, 48, 1, 13824, 1714176, 21606912, 76317696, 110160576, 78451200, 30645504, 6976512, 953424, 78400, 3760, 96, 1, 331776, 207028224, 8190885888, 74684104704, 253100173824

Offset: 1

Wolfdieter Lang, Dec 01 2003

The row length sequence for this array is [1,5,9,13,17,...] = A016813(n-1), n >= 1.
The g.f. for the k-th column, (with leading zeros and k >= 4) is G(k,x) = x^ceiling(k/4)*P(k,x)/Product_{p = 4..k} (1 - fallfac(p,4)*x), with fallfac(n,m) := A008279(n,m) (falling factorials) and P(k,x) := Sum_{m = 0..kmax(k)} A090221(k,m)*x^m, k >= 4, with kmax(k) := A057353(k-4)= floor(3*(k-4)/4). For the recurrence of the G(k,x) see A090221.
Codara et al., show that T(n,k) gives the number of k-colorings of the graph nK_4 (the disjoint union of n copies of the complete graph K_4). - Peter Bala, Aug 15 2013

			Table begins
n\k|   4      5      6      7      8     9   10   11   12
= = = = = = = = = = = = = = = = = = = = = = = = = = = = =
1  |   1
2  |  24     96     72     16      1
3  | 576  13824  50688  59904  30024  7200  856   48    1
...

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 71, flattened)
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Codara, O. M. D’Antona, and P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv:1308.1700v1 [cs.DM], 2013.
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
Wolfdieter Lang, First 4 rows.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Cf. A090215, A071379 (row sums), A090213 (alternating row sums).

S_{1, 1} = A008277, S_{2, 1} = A008297 (ignoring signs), S_{3, 1} = A035342, S_{2, 2} = A078739, S_{3, 2} = A078740, S_{3, 3} = A078741.

T:= (n,k) -> (-1)^k/k!*add((-1)^p*binomial(k,p)*(p*(p-1)*(p-2)*(p-3))^n,p=4..k):
seq(seq(T(n,k),k=4..4*n),n=1..10); # Robert Israel, Jan 28 2016

a[n_, k_] := (((-1)^k)/k!)*Sum[((-1)^p)*Binomial[k, p]*FactorialPower[p, 4]^n, {p, 4, k}]; Table[a[n, k], {n, 1, 5}, {k, 4, 4*n}] // Flatten (* Jean-François Alcover, Sep 05 2012, updated Jan 28 2016 *)

a(n, k) = (-1)^k/k! * Sum_{p = 4..k} (-1)^p * binomial(k, p) * fallfac(p, 4)^n,  with fallfac(p, 4) := A008279(p, 4) = p*(p - 1)*(p - 2)*(p - 3); 4 <= k <= 4*n, n >= 1, else 0. From eq.(19) with r = 4 of the Blasiak et al. reference.
E^n = Sum_{k = 4..4*n} a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator (x^4)*d^4/dx^4.
The row polynomials R(n,x) are given by the Dobinski-type formula R(n,x) = exp(-x)*Sum_{k >= 0} (k*(k - 1)*(k - 2)*(k - 3))^n*x^k/k!. - Peter Bala, Aug 15 2013

A010796 a(n) = n!*(n+1)!/2.

Original entry on oeis.org

1, 6, 72, 1440, 43200, 1814400, 101606400, 7315660800, 658409472000, 72425041920000, 9560105533440000, 1491376463216640000, 271430516305428480000, 57000408424139980800000, 13680098021793595392000000, 3720986661927857946624000000

Offset: 1

N. J. A. Sloane

Column 2 in triangle A009963.
a(n) = A078740(n, 2), first column of (3, 2)-Stirling2 array.
Also the number of undirected Hamiltonian paths in the complete bipartite graph K_{n,n+1}. - Eric W. Weisstein, Sep 03 2017
Also, the number of undirected Hamiltonian cycles in the complete bipartite graph K_{n+1,n+1}. - Pontus von Brömssen, Sep 06 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Index entries for sequences related to factorial numbers.

Cf. A006472, A009963, A078740, A010790.

Main diagonal of A291909.

[Factorial(n)* Factorial(n+1) / 2: n in [1..20]]; // Vincenzo Librandi, Jun 11 2013

Table[n! (n + 1)! / 2, {n, 1, 20}] (* Vincenzo Librandi, Jun 11 2013 *)
Times@@@Partition[Range[20]!,2,1]/2 (* Harvey P. Dale, Jul 04 2017 *)

for(n=1,30, print1(n!*(n+1)!/2, ", ")) \\ G. C. Greubel, Feb 07 2018

a(n) = 2^(n-1) * A006472(n+1).
a(n) = A010790(n)/2.
E.g.f.: (hypergeom([1, 2], [], x)-1)/2.
a(n) = Product_{k=1..n-1} (k^2+3*k+2). - Gerry Martens, May 09 2016
E.g.f.: x*hypergeom([1, 3], [], x). - Robert Israel, May 09 2016
From Amiram Eldar, Jun 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*(BesselI(1, 2) - 1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(1 - BesselJ(1, 2)). (End)

cp's OEIS Frontend

A090452 Scaled array A078740 ((3,2)-Stirling2).

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A090437 Alternating row sums of array A078740 ((3,2)-Stirling2).

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A091549 Second column (k=3) sequence of array A078740 ((3,2)-Stirling2) divided by 6.

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A091741 Coefficients of certain polynomials related to array A078740 ((3,2)-Stirling2).

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A078739 Triangle of generalized Stirling numbers S_{2,2}(n,k) read by rows (n>=1, 2<=k<=2n).

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A078741 Triangle of generalized Stirling numbers S_{3,3}(n,k) read by rows (n>=1, 3<=k<=3n).

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A090438 Generalized Stirling2 array (4,2).

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A091534 Generalized Stirling2 array (5,2).

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