login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000369 Triangle of numbers related to triangle A049213; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497. 17

%I #67 Mar 23 2020 12:11:31

%S 1,3,1,21,9,1,231,111,18,1,3465,1785,345,30,1,65835,35595,7650,825,45,

%T 1,1514205,848925,196245,24150,1680,63,1,40883535,23586255,5755050,

%U 775845,62790,3066,84,1,1267389585,748471185,190482705,27478710

%N Triangle of numbers related to triangle A049213; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.

%C 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).

%C 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

%C 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

%H Vincenzo Librandi, <a href="/A000369/b000369.txt">Rows n = 1..50, flattened</a>

%H P. Blasiak, K. A. Penson and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0212072, 2002.

%H Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/08/23/a-class-of-differential-operators-and-the-stirling-numbers/">A Class of Differential Operators and the Stirling Numbers</a>

%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) 09.8.3

%H Wolfdieter Lang, <a href="/A000369/a000369.txt">First ten rows.</a>

%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

%H Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Schork/schork2.html">The Generalized Stirling and Bell Numbers Revisited</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.

%H Mathias Pétréolle, Alan D. Sokal, <a href="https://arxiv.org/abs/1907.02645">Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions</a>, arXiv:1907.02645 [math.CO], 2019.

%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>

%F 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.

%F E.g.f. of m-th column: ((1-(1-4*x)^(1/4))^m)/m!.

%F From _Peter Bala_, Jun 08 2016: (Start)

%F 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! + ....

%F 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).

%F 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)

%e {1}; {3,1}; {21,9,1}; {231,111,18,1}; {3465,1785,345,30,1}; ...

%e 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

%t 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 *)

%o (Sage) # uses[bell_transform from A264428]

%o # Adds a column 1,0,0,0,... at the left side of the triangle.

%o def A000369_row(n):

%o multifact_4_3 = lambda n: prod(4*k + 3 for k in (0..n-1))

%o mfact = [multifact_4_3(k) for k in (0..n)]

%o return bell_transform(n, mfact)

%o [A000369_row(n) for n in (0..9)] # _Peter Luschny_, Dec 31 2015

%Y Row sums give A016036. Cf. A004747.

%Y Columns include A008545.

%Y Alternating row sums A132163.

%K easy,nonn,tabl

%O 1,2

%A _Wolfdieter Lang_

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 29 09:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)