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A000369
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Triangle of numbers related to triangle A049213; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
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16
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers
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.
Index entries for sequences related to Bessel functions or polynomials
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FORMULA
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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)
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EXAMPLE
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{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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Row sums give A016036. Cf. A004747.
Columns include A008545.
Alternating row sums A132163.
Sequence in context: A144280 A107717 A143173 * A225471 A136236 A113090
Adjacent sequences: A000366 A000367 A000368 * A000370 A000371 A000372
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Wolfdieter Lang
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STATUS
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approved
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