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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.

%I A203412 #40 Jul 31 2025 07:35:39
%S A203412 1,1,1,4,3,1,28,19,6,1,280,180,55,10,1,3640,2260,675,125,15,1,58240,
%T A203412 35280,10360,1925,245,21,1,1106560,658000,190680,35385,4620,434,28,1,
%U A203412 24344320,14266560,4090240,756840,100065,9828,714,36,1
%N 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.
%C A203412 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
%H A203412 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 A203412 T. Mansour, M. Schork, and M. Shattuck, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p77/0">On a new family of generalized Stirling and Bell numbers</a>, Electron. J. Combin. 18 (2011) #P77 (33 pp.).
%H A203412 Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="http://dx.doi.org/10.1016/j.aml.2012.02.009">On the Stirling numbers associated with the meromorphic Weyl algebra</a>, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771.
%F A203412 (1) Is given by the recurrence relation
%F A203412   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.
%F A203412 (2) Is given explicitly by
%F A203412   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.
%F A203412 a(n,1) = A007559(n-1). - _Peter Luschny_, Dec 21 2015
%e A203412 Triangle starts:
%e A203412 [    1]
%e A203412 [    1,     1]
%e A203412 [    4,     3,     1]
%e A203412 [   28,    19,     6,    1]
%e A203412 [  280,   180,    55,   10,   1]
%e A203412 [ 3640,  2260,   675,  125,  15,  1]
%e A203412 [58240, 35280, 10360, 1925, 245, 21, 1]
%p A203412 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
%t A203412 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 *)
%o A203412 (Sage) # uses[bell_transform from A264428]
%o A203412 triplefactorial = lambda n: prod(3*k + 1 for k in (0..n-1))
%o A203412 def A203412_row(n):
%o A203412     trifact = [triplefactorial(k) for k in (0..n)]
%o A203412     return bell_transform(n, trifact)
%o A203412 [A203412_row(n) for n in (0..8)] # _Peter Luschny_, Dec 21 2015
%Y A203412 Cf. A007559, A032031, A039683, A051141, A132062, A264428.
%K A203412 nonn,tabl
%O A203412 1,4
%A A203412 _Mark Shattuck_, Jan 01 2012