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A035469 Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.

%I A035469 #99 Jul 31 2025 07:34:37
%S A035469 1,4,1,28,12,1,280,160,24,1,3640,2520,520,40,1,58240,46480,11880,1280,
%T A035469 60,1,1106560,987840,295960,40040,2660,84,1,24344320,23826880,8090880,
%U A035469 1296960,109200,4928,112,1,608608000,643843200
%N A035469 Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.
%C A035469 Previous name was: Triangle of numbers related to triangle A035529; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297 and A035342.
%C A035469 a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing quartic (4-ary) trees. Proof based on the a(n,m) recurrence. See a D. Callan comment on the m=1 case A007559. See also the F. Bergeron et al. reference, especially Table 1, first row and Example 1 for the e.g.f. for m=1. - _Wolfdieter Lang_, Sep 14 2007
%C A035469 For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 19 2016
%D A035469 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.
%H A035469 Peter Bala, <a href="/A035342/a035342_Bala.txt">Generalized Dobinski formulas</a>
%H A035469 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/0402027, 2004.
%H A035469 Richell O. Celeste, Roberto B. Corcino and Ken Joffaniel M. Gonzales. <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Celeste/celeste3.html">Two Approaches to Normal Order Coefficients</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
%H A035469 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 A035469 Tom Copeland, <a href="http://tcjpn.wordpress.com/">Mathemagical Forests</a>
%H A035469 Tom Copeland, <a href="http://tcjpn.wordpress.com/">Addendum to Mathemagical Forests</a>
%H A035469 Milan 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 A035469 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 A035469 Wolfdieter Lang, <a href="/A035469/a035469.txt">First 10 rows</a>.
%H A035469 Shi-Mei Ma, <a href="http://arxiv.org/abs/1208.3104">Some combinatorial sequences associated with context-free grammars</a>, arXiv:1208.3104 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
%H A035469 E. Neuwirth, <a href="http://homepage.univie.ac.at/erich.neuwirth/papers/TechRep99-05.pdf">Recursively defined combinatorial functions: Extending Galton's board</a>, Discrete Math. 239 (2001) 33-51.
%H A035469 Mathias Pétréolle and 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.
%F A035469 a(n, m) = Sum_{j=m..n} |A051141(n, j)|*S2(j, m) (matrix product), with S2(j, m):=A008277(j, m) (Stirling2 triangle). Priv. comm. to _Wolfdieter Lang_ by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.
%F A035469 a(n, m) = n!*A035529(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;
%F A035469 E.g.f. of m-th column: ((-1+(1-3*x)^(-1/3))^m)/m!.
%F A035469 From _Peter Bala_, Nov 25 2011: (Start)
%F A035469 E.g.f.: G(x,t) = exp(t*A(x)) = 1 + t*x + (4*t+t^2)*x^2/2! + (28*t + 12*t^2 + t^3)*x^3/3! + ..., where A(x) = -1 + (1-3*x)^(-1/3) satisfies the autonomous differential equation A'(x) = (1+A(x))^4.
%F A035469 The generating function G(x,t) satisfies the partial differential equation t*(dG/dt+G) = (1-3*x)*dG/dx, from which follows the recurrence given above.
%F A035469 The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^4*d/dx. Cf. A008277 (D = (1+x)*d/dx), A105278 (D = (1+x)^2*d/dx), A035342 (D = (1+x)^3*d/dx) and A049029 (D = (1+x)^5*d/dx).
%F A035469 (End)
%F A035469 Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k>=0} k*(k+3)*(k+6)*...*(k+3*(n-1))*x^k/k!. - _Peter Bala_, Jun 23 2014
%e A035469 Triangle starts:
%e A035469      {1}
%e A035469      {4,    1}
%e A035469     {28,   12,    1}
%e A035469    {280,  160,   24,    1}
%e A035469   {3640, 2520,  520,   40,    1}
%t A035469 a[n_, m_] /; n >= m >= 1 := a[n, m] = (3(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 *)
%t A035469 rows = 9;
%t A035469 a[n_, m_] := BellY[n, m, Table[Product[3k+1, {k, 0, j}], {j, 0, rows}]];
%t A035469 Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018 *)
%o A035469 (Sage) # uses[bell_matrix from A264428]
%o A035469 # Adds a column 1,0,0,0, ... at the left side of the triangle.
%o A035469 bell_matrix(lambda n: A007559(n+1) , 9) # _Peter Luschny_, Jan 19 2016
%Y A035469 a(n, m)=: S2(4, n, m) is the fourth triangle of numbers in the sequence S2(1, n, m) := A008277(n, m) (Stirling 2nd kind), S2(2, n, m) := A008297(n, m) (Lah), S2(3, n, m) := A035342(n, m). a(n, 1)= A007559(n).
%Y A035469 Row sums: A049119(n), n >= 1.
%Y A035469 Cf. A094638.
%K A035469 easy,nice,nonn,tabl
%O A035469 1,2
%A A035469 _Wolfdieter Lang_
%E A035469 New name from _Peter Luschny_, Jan 19 2016