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

1, 4, 1, 28, 12, 1, 280, 160, 24, 1, 3640, 2520, 520, 40, 1, 58240, 46480, 11880, 1280, 60, 1, 1106560, 987840, 295960, 40040, 2660, 84, 1, 24344320, 23826880, 8090880, 1296960, 109200, 4928, 112, 1, 608608000, 643843200

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nice
• nonn
• tabl

Comments

Previous name was: Triangle of numbers related to triangle A035529; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297 and A035342.

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

For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

Examples

			Triangle starts:
     {1}
     {4,    1}
    {28,   12,    1}
   {280,  160,   24,    1}
  {3640, 2520,  520,   40,    1}
		

References

• 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.

Links

• Peter Bala, Generalized Dobinski formulas
• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
• Richell O. Celeste, Roberto B. Corcino and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
• Tom Copeland, A Class of Differential Operators and the Stirling Numbers
• Tom Copeland, Mathemagical Forests
• Tom Copeland, Addendum to Mathemagical Forests
• Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
• Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
• Wolfdieter Lang, First 10 rows.
• Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
• E. Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 (2001) 33-51.
• Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.

Crossrefs

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).

Row sums: A049119(n), n >= 1.

Cf. A094638.

Programs

Mathematica

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 *)
rows = 9;
a[n_, m_] := BellY[n, m, Table[Product[3k+1, {k, 0, j}], {j, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

Sage

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: A007559(n+1) , 9) # Peter Luschny, Jan 19 2016

Formula

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.

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;

E.g.f. of m-th column: ((-1+(1-3*x)^(-1/3))^m)/m!.

From Peter Bala, Nov 25 2011: (Start)

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.

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.

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).

(End)

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

Extensions

New name from Peter Luschny, Jan 19 2016