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. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+3) (A008545) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015
LINKS
Vincenzo Librandi, Rows n = 1..50, flattened
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0212072, 2002.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
Wolfdieter Lang, First ten rows.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
Mathias Pétréolle, Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
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!.
From Peter Bala, Jun 08 2016: (Start)
With offset 0, the e.g.f. is 1/(1 - 4*x)^(3/4)*exp(t*(1 - (1 - 4*x)^(1/4))) = 1 + (3 + t)*x + (21 + 9*t + t^2)*x^2/2! + ....
Thus with row and column numbering starting at 0, this triangle is the exponential Riordan array [d/dx(F(x)), F(x)], belonging to the Derivative subgroup of the exponential Riordan group, where F(x) = 1 - (1 - 4*x)^(1/4).
Row polynomial recurrence: R(n+1,t) = t*Sum_{k = 0..n} binomial(n,k)*A008545(k)*R(n-k,t) with R(0,t) = 1. (End)
EXAMPLE
Triangle begins:
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. - Wolfdieter 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}]] (* Jean-François Alcover, Jul 22 2011 *)
PROG
(Sage) # uses[bell_transform from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
def A000369_row(n):
multifact_4_3 = lambda n: prod(4*k + 3 for k in (0..n-1))
mfact = [multifact_4_3(k) for k in (0..n)]
return bell_transform(n, mfact)
[A000369_row(n) for n in (0..9)] # Peter Luschny, Dec 31 2015
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved