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A049385
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Triangle of numbers related to triangle A049375; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297...
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10
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1, 6, 1, 66, 18, 1, 1056, 372, 36, 1, 22176, 9240, 1200, 60, 1, 576576, 271656, 42840, 2940, 90, 1, 17873856, 9269568, 1685376, 142800, 6090, 126, 1, 643458816, 360847872, 73313856, 7254576, 386400, 11256, 168, 1, 26381811456, 15799069440
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n,m) := S2(6; n,m) is the sixth triangle of numbers in the sequence S2(k; n,m), k=1..6: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, respectively. a(n,1)= A008548(n).
a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 6-ary trees. Proof based on the a(n,m) recurrence. See also the F. Bergeron et al. reference, especially Table 1, first row and Example 1 for the e.g.f. for m=1. W. Lang, Sept 14 2007.
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REFERENCES
| E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's board, Discr. Math. 239 (2001) 33-51.
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.
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LINKS
| P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First 10 rows.
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FORMULA
| a(n, m) = n!*A049375(n, m)/(m!*5^(n-m)); a(n+1, m) = (5*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1; E.g.f. of m-th column: ((-1+(1-5*x)^(-1/5))^m)/m!.
a(n, m) = sum(|A051150(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. to W. Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.
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EXAMPLE
| {1}; {6,1}; {66,18,1}; {1056,372,36,1}; ...
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MATHEMATICA
| a[n_, m_] := n!*Coefficient[Series[((-1 + (1 - 5*x)^(-1/5))^m)/m!, {x, 0, n}], x^n];
Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 38]]
(* From Jean-François Alcover, Jun 21 2011, after e.g.f. *)
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CROSSREFS
| Cf. A049412.
Sequence in context: A134279 A134280 A134278 * A009384 A051151 A009330
Adjacent sequences: A049382 A049383 A049384 * A049386 A049387 A049388
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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