OFFSET
1,2
COMMENTS
Previous name was: Triangle of numbers related to triangle A049224; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
T(n, m) = S2p(-5; 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) = A008543(n-1).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016
LINKS
G. C. Greubel, Rows n = 1..50 of the triangle, flattened
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
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.
Peter Luschny, The Bell transform
FORMULA
EXAMPLE
Triangle begins as:
1;
5, 1;
55, 15, 1;
935, 295, 30, 1;
21505, 7425, 925, 50, 1;
623645, 229405, 32400, 2225, 75, 1;
21827575, 8423415, 1298605, 103600, 4550, 105, 1;
894930575, 358764175, 59069010, 5235405, 271950, 8330, 140, 1;
MATHEMATICA
(* First program *)
rows = 10;
b[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, rows}]];
A = Table[b[n, m], {n, 1, rows}, {m, 1, rows}] // Inverse // Abs;
A013988 = Table[A[[n, m]], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==0, 0, If[k==n, 1, (6*(n-1) -k)*T[n-1, k] +T[n-1, k-1]]];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Oct 03 2023 *)
PROG
(Sage) # uses[inverse_bell_matrix from A264428]
# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
inverse_bell_matrix(lambda n: factorial(n)*binomial(5, n), 8) # Peter Luschny, Jan 16 2016
(Magma)
function T(n, k) // T = A013988
if k eq 0 then return 0;
elif k eq n then return 1;
else return (6*(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, Jan 16 2016
STATUS
approved