|
| |
|
|
A000369
|
|
Triangle of numbers related to triangle A049213; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
|
|
15
| |
|
|
1, 3, 1, 21, 9, 1, 231, 111, 18, 1, 3465, 1785, 345, 30, 1, 65835, 35595, 7650, 825, 45, 1, 1514205, 848925, 196245, 24150, 1680, 63, 1, 40883535, 23586255, 5755050, 775845, 62790, 3066, 84, 1, 1267389585, 748471185, 190482705, 27478710
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| a(n,m) := S2p(-3; 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)= A008545(n-1).
a(n,m), n>=m>=1, enumerates unordered n-vertex m-forests composed of m increasing plane (aka ordered) trees, with one vertex of out-degree r=0 (leafs or a root) and each vertex with out-degree r>=1 comes in r+2 types (like for an (r+2)-ary vertex). Proof from the e.g.f. of the first column Y(z):=1-(1-4*x)^(1/4) and the F. Bergeron et al. reference given in A001498, eq. (8), Y'(z)= phi(Y(z)), Y(0)=0, with out-degree o.g.f. phi(w)=1/(1-w)^3. W. Lang Oct 12 2007.
|
|
|
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.
W. Lang, First ten rows.
|
|
|
FORMULA
| a(n, m) = n!*A049213(n, m)/(m!*4^(n-m)); a(n+1, m) = (4*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-(1-4*x)^(1/4))^m)/m!.
|
|
|
EXAMPLE
| {1}; {3,1}; {21,9,1}; {231,111,18,1}; {3465,1785,345,30,1}; ...
Tree combinatorics for a(3,2)=9: there are three m=2 forests each with one tree a root
(with out-degree r=0) and the other tree a root and a leaf coming in three versions (like for a 3-ary vertex). Each such forest can be labeled increasingly in three ways (like (1,(23)), (2,(13)) and (3,(12)) yielding 9 such forests. W. Lang Oct 12 2007.
|
|
|
MATHEMATICA
| a[n_, m_] /; n >= m >= 1 := a[n, m] = (4(n-1) - m)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* From Jean-François Alcover, Jul 22 2011 *)
|
|
|
CROSSREFS
| Row sums give A016036. Cf. A004747.
Columns include A008545.
Alternating row sums A132163.
Sequence in context: A144280 A107717 A143173 * A136236 A113090 A138354
Adjacent sequences: A000366 A000367 A000368 * A000370 A000371 A000372
|
|
|
KEYWORD
| easy,nonn,tabl
|
|
|
AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
|
| |
|
|
