OFFSET
1,2
COMMENTS
Previous name was: Triangle of numbers related to triangle A048966; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
T(n,m) = S2p(-2; 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). T(n,1)= A008544(n-1).
T(n,m), n>=m>=1, enumerates unordered n-vertex m-forests composed of m plane (aka ordered) increasing (rooted) trees where vertices of out-degree r>=0 come in r+1 different types (like an (r+1)-ary vertex). Proof from the e.g.f. of the first column Y(z) = 1 - (1-3*x)^(1/3) and the F. Bergeron et al. eq. (8) Y'(z)= phi(Y(z)), Y(0) = 0, with out-degree o.g.f. phi(w)=1/(1-w)^2. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of the triple factorial numbers A008544 which adds a first column (1,0,0 ...) on the left side of the triangle. For the definition of the Bell transform see A264428. See A051141 for the triple factorial numbers A032031 and A203412 for the triple factorial numbers A007559 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 21 2015
Let M be the infinite lower unit triangular array A136216. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0 M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to the present triangle. See the Example section. - Peter Bala, Nov 09 2025
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Peter Bala, Generalized Dobinski formulas.
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of increasing trees, 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.
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.
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, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) #09.3.3.
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.
FORMULA
T(n, m) = n!*A048966(n, m)/(m!*3^(n-m));
T(n+1, m) = (3*n-m)*T(n, m)+ T(n, m-1), for n >= m >= 1, with T(n, m) = 0, for n<m, and T(n, 0) = 0, T(1, 1) = 1.
E.g.f. of m-th column: ( 1 - (1-3*x)^(1/3) )^m/m!.
Sum_{k=1..n} T(n, k) = A015735(n).
For a formula expressed as special values of hypergeometric functions 3F2 see the Maple program below. - Karol A. Penson, Feb 06 2004
T(n,1) = A008544(n-1). - Peter Luschny, Dec 23 2015
Dobinski_type formula: the n-th row polynomial R(n, x) = exp(x)*Sum_{k >= 0} (-1)^(n+k)*k*(k - 3)*(k - 6)*...*(k - 3*(n-1))*x^k/k!. For example, R(4, x) = exp(x)*Sum_{k >= 0} (-1)^(4+k)*k*(k - 3)*(k - 6)*(k - 9)*x^k/k! = 80*x + 52*x^2 + 12*x^3 + x^4. - Peter Bala, Nov 08 2025
EXAMPLE
Triangle begins:
1;
2, 1;
10, 6, 1;
80, 52, 12, 1;
880, 600, 160, 20, 1;
12320, 8680, 2520, 380, 30, 1;
209440, 151200, 46480, 7840, 770, 42, 1;
Tree combinatorics for T(3,2)=6: Consider first the unordered forest of m=2 plane trees with n=3 vertices, namely one vertex with out-degree r=0 (root) and two different trees with two vertices (one root with out-degree r=1 and a leaf with r=0). The 6 increasing labelings come then from the forest with rooted (x) trees x, o-x (1,(3,2)), (2,(3,1)) and (3,(2,1)) and similarly from the second forest x, x-o (1,(2,3)), (2,(1,3)) and (3,(1,2)).
From Peter Bala, Nov 09 2025: (Start)
With the array M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/ 1 \ /1 \ /1 \ / 1 \
| 2 1 ||0 1 ||0 1 | | 2 1 |
|10 4 1 ||0 2 1 ||0 0 1 |... = |10 6 1 |
|80 30 6 1 ||0 10 4 1 ||0 0 2 1 | |80 52 12 1|
|... ||0 80 30 6 1 ||0 0 10 4 1| |... |
|... ||... ||... | | |
(End)
MAPLE
T := (n, m) -> 3^n/m!*(1/3*m*GAMMA(n-1/3)*hypergeom([1-1/3*m, 2/3-1/3*m, 1/3-1/3*m], [2/3, 4/3-n], 1)/GAMMA(2/3)-1/6*m*(m-1)*GAMMA(n-2/3)*hypergeom( [1-1/3*m, 2/3-1/3*m, 4/3-1/3*m], [4/3, 5/3-n], 1)/Pi*3^(1/2)*GAMMA(2/3)):
for n from 1 to 6 do seq(simplify(T(n, k)), k=1..n) od;
# Karol A. Penson, Feb 06 2004
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> mul(3*k+2, k=(0..n-1)), 9); # Peter Luschny, Jan 29 2016
MATHEMATICA
T[1, 1]= 1; T[_, 0]= 0; T[0, _]= 0; T[n_, m_]:= (3*(n-1)-m)*T[n-1, m]+T[n-1, m-1];
Flatten[Table[T[n, m], {n, 12}, {m, n}] ][[1 ;; 45]] (* Jean-François Alcover, Jun 16 2011, after recurrence *)
(* Alternative: *)
f[n_, m_]:= m/n Sum[Binomial[k, n-m-k] 3^k (-1)^(n-m-k) Binomial[n+k-1, n-1], {k, 0, n-m}]; Table[n! f[n, m]/(m! 3^(n-m)), {n, 12}, {m, n}]//Flatten (* Michael De Vlieger, Dec 23 2015 *)
(* Alternative: *)
rows = 12;
T[n_, m_]:= BellY[n, m, Table[Product[3k+2, {k, 0, j-1}], {j, 0, rows}]];
Table[T[n, m], {n, rows}, {m, n}]//Flatten (* Jean-François Alcover, Jun 22 2018 *)
PROG
(SageMath) # uses [bell_transform from A264428]
triplefactorial = lambda n: prod(3*k+2 for k in (0..n-1))
def A004747_row(n):
trifact = [triplefactorial(k) for k in (0..n)]
return bell_transform(n, trifact)
[A004747_row(n) for n in (0..10)] # Peter Luschny, Dec 21 2015
(Magma)
function T(n, k) // T = A004747
if k eq 0 then return 0;
elif k eq n then return 1;
else return (3*(n-1)-k)*T(n-1, k) + T(n-1, k-1);
end if;
end function;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2023
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
New name from Peter Luschny, Dec 21 2015
STATUS
approved