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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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10
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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 (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Dec 01 2003
For the computation of the k-th column sequence see A090219.
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REFERENCES
| P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
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LINKS
| 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.
W. Lang, First 6 rows.
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FORMULA
| a(n, k)= (((-1)^k)/k!)*sum(((-1)^p)* binomial(k, p)*fallfac(p, 3)^n, p=3..k), 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}^(3n) a(n,k)*x^k*D^k where D is the operator d/dx, and E the operator x^3d^3/dx^3.
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EXAMPLE
| 1; 6,18,9,1; 36,540,1242,882,243,27,1; ...
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CROSSREFS
| 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).
Sequence in context: A077022 A074923 A093061 * A129870 A091014 A097370
Adjacent sequences: A078738 A078739 A078740 * A078742 A078743 A078744
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KEYWORD
| nonn,tabf,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 21 2002
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