A078741 - OEIS

%I #40 Oct 16 2023 11:28:04

%S 1,6,18,9,1,36,540,1242,882,243,27,1,216,13608,94284,186876,149580,

%T 56808,11025,1107,54,1,1296,330480,6148872,28245672,49658508,41392620,

%U 18428400,4691412,706833,63375,3285,90,1,7776,7954848,380841264,3762380016,13062960720

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

%C The sequence of row lengths for this array is [1,4,7,10,..]= A016777(n-1), n>=1.

%C 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

%C For the computation of the k-th column sequence see A090219.

%C 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

%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>

%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027,2004.

%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://dx.doi.org/10.1016/S0375-9601(03)00194-4">The general boson normal ordering problem</a>, Phys. Lett. A 309 (2003) 198-205.

%H P. Codara, O. M. D’Antona, P. Hell, <a href="http://arxiv.org/abs/1308.1700">A simple combinatorial interpretation of certain generalized Bell and Stirling numbers</a> arXiv:1308.1700v1 [cs.DM]

%H A. Dzhumadildaev and D. Yeliussizov, <a href="http://arxiv.org/abs/1408.6764v1">Path decompositions of digraphs and their applications to Weyl algebra</a>, 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]

%H Askar Dzhumadil’daev and Damir Yeliussizov, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p10">Walks, partitions, and normal ordering</a>, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.

%H W. Lang, <a href="/A078741/a078741.txt">First 6 rows</a>.

%F 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.

%F 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.

%F 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

%e From _Peter Bala_, Aug 15 2013: (Start)

%e The table begins

%e n\k |   3     4     5      6      7     8     9   10  11  12

%e = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

%e   1 |   1

%e   2 |   6    18     9      1

%e   3 |  36   540  1242    882    243    27     1

%e   4 | 216 13608 94284 186876 149580 56808 11025 1107  54   1

%e ...

%e Graph coloring interpretation of T(2,3) = 6:

%e The graph 2K_3 is 2 copies of K_3, the complete graph on 3 vertices:

%e     o b      o e

%e    / \      / \

%e   o---o    o---o

%e   a   c    d   f

%e 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)

%t 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 *)

%Y Row sums give A069223. Cf. A078739.

%Y The column sequences (without leading zeros) are A000400 (powers of 6), 18*A089507, 9*A089518, A089519, etc.

%Y A089504, A069223 (row sums), A090212 (alternating row sums).

%Y A078740, A090214.

%K nonn,tabf,easy

%O 1,2

%A _N. J. A. Sloane_, Dec 21 2002