A203412 Triangle read by rows, a(n,k), n>=k>=1, which represent the s=3, h=1 case of a two-parameter generalization of Stirling numbers arising in conjunction with normal ordering.

1, 1, 1, 4, 3, 1, 28, 19, 6, 1, 280, 180, 55, 10, 1, 3640, 2260, 675, 125, 15, 1, 58240, 35280, 10360, 1925, 245, 21, 1, 1106560, 658000, 190680, 35385, 4620, 434, 28, 1, 24344320, 14266560, 4090240, 756840, 100065, 9828, 714, 36, 1

Offset

1,4

Comments

Also the Bell transform of the triple factorial numbers A007559 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 A004747 for the triple factorial numbers A008544 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 23 2015

Links

Table of n, a(n) for n=1..45.

Richell O. Celeste, Roberto B. Corcino, Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.

T. Mansour, M. Schork, and M. Shattuck, On a new family of generalized Stirling and Bell numbers, Electron. J. Combin. 18 (2011) #P77 (33 pp.).

Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771.

Formula

(1) Is given by the recurrence relation

a(n+1,k) = a(n,k-1)+(3*n-2*k)*a(n,k) if n>=0 and k>=1, along with the initial values a(n,0) = delta_{n,0} and a(0,k) = delta_{0,k} for all n,k>=0.

(2) Is given explicitly by

a(n,k) = (n!*3^n)/(k!*2^k)*Sum{j=0..k} (-1)^j*C(k,j)*C(n-2*j/3-1,n) for all n>=k>=1.

a(n,1) = A007559(n-1). - Peter Luschny, Dec 21 2015

Example

Triangle starts:
[    1]
[    1,     1]
[    4,     3,     1]
[   28,    19,     6,    1]
[  280,   180,    55,   10,   1]
[ 3640,  2260,   675,  125,  15,  1]
[58240, 35280, 10360, 1925, 245, 21, 1]

Maple

A203412 := (n, k) -> (n!*3^n)/(k!*2^k)*add((-1)^j*binomial(k, j)*binomial(n-2*j/3-1, n), j=0..k): seq(seq(A203412(n, k), k=1..n), n=1..9); # Peter Luschny, Dec 21 2015

Mathematica

Table[(n! 3^n)/(k! 2^k) Sum[ (-1)^j Binomial[k, j] Binomial[n - 2 j/3 - 1, n], {j, 0, k}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Dec 23 2015 *)

Prog

(Sage) # uses[bell_transform from A264428]
triplefactorial = lambda n: prod(3*k + 1 for k in (0..n-1))
def A203412_row(n):
    trifact = [triplefactorial(k) for k in (0..n)]
    return bell_transform(n, trifact)
[A203412_row(n) for n in (0..8)] # Peter Luschny, Dec 21 2015

Crossrefs

Cf. A007559, A032031, A039683, A051141, A132062, A264428.

Sequence in context: A245732 A039621 A142158 * A217756 A154960 A143543

Adjacent sequences: A203409 A203410 A203411 * A203413 A203414 A203415

Keyword

nonn,tabl

Author

Mark Shattuck, Jan 01 2012

Status

approved