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A004747
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Triangle of numbers related to triangle A048966; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
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12
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1, 2, 1, 10, 6, 1, 80, 52, 12, 1, 880, 600, 160, 20, 1, 12320, 8680, 2520, 380, 30, 1, 209440, 151200, 46480, 7840, 770, 42, 1, 4188800, 3082240, 987840, 179760, 20160, 1400, 56, 1, 96342400, 71998080, 23826880, 4583040, 562800, 45360, 2352, 72, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n,m) := S2p(-2; n,m), a member of a sequence of triangles including S2p(-1; n,m) := A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1; n,m) := A008277(n,m) (Stirling 2nd kind). a(n,1)= A008544(n-1).
a(n,m), n>=m>=1, enumerates unordered n-vertex m-forests composed of m plane (aka ordered) increasing (rooted) trees where vertices of out-degree r>=0 come in r+1 different types (like an (r+1)-ary vertex). Proof from the e.g.f. of the first column Y(z):=1-(1-3*x)^(1/3) and the F. Bergeron et al. eq. (8) Y'(z)= phi(Y(z)), Y(0)=0, with out-degree o.g.f. phi(w)=1/(1-w)^2. W. Lang Oct 12 2007.
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REFERENCES
| 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
| W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Index entries for sequences related to Bessel functions or polynomials
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
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FORMULA
| a(n, m) = n!*A048966(n, m)/(m!*3^(n-m)); a(n+1, m) = (3*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-(1-3*x)^(1/3))^m)/m!.
Formula: expressed as special values of hypergeometric functions 3F2, in Maple notation: a(n, m)=3^n/m!*(1/3*m*GAMMA(n-1/3)*hypergeom([1-1/3*m, 2/3-1/3*m, 1/3-1/3*m], [2/3, 4/3-n], 1)/GAMMA(2/3)-1/6*m*(m-1)*GAMMA(n-2/3) *hypergeom([1-1/3*m, 2/3-1/3*m, 4/3-1/3*m], [4/3, 5/3-n], 1)/Pi*3^(1/2)*GAMMA(2/3)). From Karol A. Penson ( penson(AT)lptl.jussieu.fr ), Feb 6 2004.
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EXAMPLE
| Triangle begins:
1;
2,1;
10,6,1;
80,52,12,1;
880,600,160,20,1; ...
Tree combinatorics for a(3,2)=6: Consider first the unordered forest of m=2 plane trees with n=3 vertices, namely one vertex with out-degree r=0 (root) and two different trees with two vertices (one root with out-degree r=1 and a leaf with r=0). The 6 increasing labelings come then from the forest with rooted (x) trees x, o-x (1,(3,2)), (2,(3,1)) and (3,(2,1)) and similarly from the second forest x, x-o (1,(2,3)), (2,(1,3)) and (3,(1,2)).
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MATHEMATICA
| a[1, 1] = 1; a[_, 0] = 0; a[0, _] = 0;
a[n_, m_] := (3*(n-1) - m)*a[n-1, m] + a[n-1, m-1];
Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}] ][[1 ;; 45]]
(* From Jean-François Alcover, Jun 16 2011, after recurrence *)
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CROSSREFS
| Row sums give A015735.
Sequence in context: A066868 A193900 A143172 * A155810 A081099 A122017
Adjacent sequences: A004744 A004745 A004746 * A004748 A004749 A004750
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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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