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A078741
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Triangle of generalized Stirling numbers S_{3,3}(n,k) read by rows (n>=1, 3<=k<=3n).
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17
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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)
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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STATUS
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approved
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