A089518 -id:A089518 - cp's OEIS frontend

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

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

A090219 Signed triangle used to compute column sequences of array A078741 ((3,3)-Stirling2).

Original entry on oeis.org

1, -1, 4, 1, -8, 10, -1, 12, -30, 20, 1, -64, 600, -1600, 1225, -1, 80, -1000, 4000, -6125, 3136, 1, -96, 1500, -8000, 18375, -18816, 7056, -1, 448, -21000, 280000, -1500625, 3687936, -4148928, 1728000, 1, -512, 28000, -448000, 3001250, -9834496, 16595712, -13824000, 4492125, -1

Offset: 3

Wolfdieter Lang, Dec 01 2003

The formula for the column no. k sequence of array A078741 is c(k;n) = b(k-2)*sum(a(k,m)*fallfac(m+2,3)^n,m=1..k-2),n>=0, k>=3 and fallfac(p,3) and b(n) are defined in the formula below.

			The third (k=5) column sequence of array A078741 is A078741(n+3,5)=c(5; n)= b(3)*(1*(3*2*1)^n -8*(4*3*2)^n +10*(5*4*3)^n), with b(3)= N(3)/A090220(3)=3/1=3, n>=0. This is 9*A089518.
The fifth (k=7) column sequence of array A078741 is A078741(n+3,7)=c(7; n)= b(5)*(1*(3*2*1)^n -64*(4*3*2)^n +600*(5*4*3)^n -1600*(6*5*4)^n +1225*(7*6*5)^n), with b(5)= N(5)/A090220(5)=3/2, n>=0. This is the sequence [243, 149580, 49658508, 13062960720,... ] which has a factor of 27.
Triangle begins:
  [1];
  [-1,4];
  [1,-8,10];
  [-1,12,-30,20];
  [1,-64,600,-1600,1225];
  ...

Wolfdieter Lang, First 10 rows.

Companion sequence A090220 for denominators D(m).

a(n, m) = A089505(n-2, m)*(sum(A089517(n, p)/fallfac(m+2, 3)^p, p=0..floor(2*(n-3)/3)))/b(n-2), n>=3, 1<= m<= n-2, else 0; with fallfac(q, 3)=A008279(q, 3)=q*(q-1)*(q-2) and b(n)=N(n)/D(n) where D(n) := A090220(n) and N(n) is given in A090220 for n=1..26.

A091553 Third column (k=6) sequence of array A090214 ((4,4)-Stirling2) divided by 72.

Original entry on oeis.org

1, 704, 300096, 113762304, 41644855296, 15075073327104, 5436979231850496, 1958506906364411904, 705205813266345885696, 253891292037560301256704, 91402929045514567230160896, 32905302125838589613523861504

Offset: 0

Wolfdieter Lang, Feb 13 2004

Cf. A089518 (third column of array (3, 3)-Stirling2 divided by 9).

a(n)= A090214(n+2, 6)/72, n>=0.
a(n)= (15*(6*5*4*3)^n - 10*(5*4*3*2)^n + (4*3*2*1)^n)/3!.
G.f.: (1+200*x)/product(1-fallfac(p, 4)*x, p=4..6), with fallfac(n, m) := A008279(n, m) (falling factorials).
a(n)= ((4!)^n)*(1-2*5^(n+1)+binomial(6, 2)^(n+1))/3!. From eq.12 of the Blasiak et al. reference given in A078740 with r=4=s, k=6.

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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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A090219 Signed triangle used to compute column sequences of array A078741 ((3,3)-Stirling2).

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A091553 Third column (k=6) sequence of array A090214 ((4,4)-Stirling2) divided by 72.

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