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A090214
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Generalized Stirling2 array S_{4,4}(n,k).
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9
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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
...
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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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.
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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