A049029 Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.

1, 5, 1, 45, 15, 1, 585, 255, 30, 1, 9945, 5175, 825, 50, 1, 208845, 123795, 24150, 2025, 75, 1, 5221125, 3427515, 775845, 80850, 4200, 105, 1, 151412625, 108046575, 27478710, 3363045, 219450, 7770, 140, 1, 4996616625, 3824996175, 1069801425

Offset

1,2

Comments

Previous name was: Triangle of numbers related to triangle A048882; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...

a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing quintic (5-ary) trees. Proof based on the a(n,m) recurrence. 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

Also the Bell transform of A007696(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

Links

Table of n, a(n) for n=1..39.

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.

T. Copeland, Mathemagical Forests

T. Copeland, Addendum to Mathemagical Forests

T. Copeland, A Class of Differential Operators and the Stirling Numbers

M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) #09.8.3.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

W. Lang, First 10 rows.

Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO]. - From N. J. A. Sloane, Aug 21 2012

E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's board, Discr. Maths. 239 (2001) 33-51.

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!*A048882(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!.

a(n, m) = sum(|A051142(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. to W. 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.

From Peter Bala, Nov 25 2011: (Start)

E.g.f.: G(x,t) = exp(t*A(x)) = 1+t*x+(5*t+t^2)*x^2/2!+(45*t+15*t^2+t^3)*x^3/3!+..., where A(x) = -1+(1-4*x)^(-1/4) satisfies the autonomous differential equation A'(x) = (1+A(x))^5.

The generating function G(x,t) satisfies the partial differential equation t*(dG/dt+G) = (1-4*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)^5*d/dx. Cf. A008277 (D = (1+x)*d/dx), A105278 (D = (1+x)^2*d/dx), A035342 (D = (1+x)^3*d/dx) and A035469 (D = (1+x)^4*d/dx).

(End)

Example

Triangle starts:
{1};
{5,1};
{45,15,1};
{585,255,30,1};
{9945,5175,825,50,1};
...

Maple

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> mul(4*k+1, k=0..n), 9); # Peter Luschny, Jan 28 2016

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 *)
rows = 9;
a[n_, m_] := BellY[n, m, Table[Product[4k+1, {k, 0, j}], {j, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

Crossrefs

a(n, m) := S2(5, n, m) is the fifth 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), S2(4, n, m) := A035469(n, m). a(n, 1)= A007696(n). A007559(n).

Cf. A048882, A007696. Row sums: A049120(n), n >= 1.

Cf. A094638

Sequence in context: A297899 A134273 A048897 * A358112 A051150 A144341

Adjacent sequences: A049026 A049027 A049028 * A049030 A049031 A049032

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang

Extensions

New name from Peter Luschny, Jan 30 2016

Status

approved