A078739 - OEIS

%I #46 Jan 30 2020 18:07:41

%S 1,2,4,1,4,32,38,12,1,8,208,652,576,188,24,1,16,1280,9080,16944,12052,

%T 3840,580,40,1,32,7744,116656,412800,540080,322848,98292,16000,1390,

%U 60,1,64,46592,1446368,9196992,20447056,20453376,10564304,3047520,511392,50400

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

%C A generalization of the Stirling2 numbers S_{1,1} from A008277.

%C 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

%C The formula for the k-th column sequence is given in A089511.

%C 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

%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>, arXiv:quant-ph/0212072, 2002.

%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 Steve Butler, Fan Chung, Jay Cummings, R. L. Graham, <a href="http://arxiv.org/abs/1504.01426">Juggling card sequences</a>, arXiv:1504.01426 [math.CO], 2015.

%H Leonard Carlitz, <a href="https://www.jstor.org/stable/2371100">On Arrays of Numbers</a>, 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 ]

%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 S.-M. Ma, T. Mansour, M. Schork.  <a href="http://arxiv.org/abs/1308.0169">Normal ordering problem and the extensions of the Stirling grammar</a>, arXiv preprint arXiv:1308.0169 [math.CO], 2013.

%H Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Schork/schork2.html">The Generalized Stirling and Bell Numbers Revisited</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.

%F 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.

%F 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.

%F 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

%F a(n, k) = A122193(n,k)*2^n/k! - Peter Luschny, Mar 25 2011

%F 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.

%F 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

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

%e The table begins

%e n\k | 2    3    4    5    6   7   8

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

%e   1 | 1

%e   2 | 2    4    1

%e   3 | 4   32   38   12    1

%e   4 | 8  208  652  576  188  24   1

%e ...

%e Graph coloring interpretation of T(2,3) = 4: The graph 2K_2 is 2 copies of K_2, the complete graph on 2 vertices:

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

%e a   b  c   d

%e The four 3-colorings of 2K_2 are ac|b|d, ad|b|c, bc|a|d and bd|a|c. (End)

%p # Note that the function implements the full triangle because it can be

%p # much better reused and referenced in this form.

%p A078739 := proc(n,k) local r;

%p add((-1)^(n-r)*binomial(n,r)*combinat[stirling2](n+r,k),r=0..n) end:

%p # Displays the truncated triangle from the definition:

%p seq(print(seq(A078739(n,k),k=2..2*n)),n=1..6); # _Peter Luschny_, Mar 25 2011

%t 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_ *)

%Y 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}).

%Y The column sequences are A000079(n-1)(powers of 2), 4*A016129(n-2), A089271, 12*A089272, A089273, etc.

%Y Main diagonal is A217900.

%Y Cf. A071951 (Legendre-Stirling triangle).

%Y Cf. A122193, A055203.

%K nonn,tabf,easy

%O 1,2

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

%E More terms from _Wolfdieter Lang_, Nov 07 2003