OFFSET
1,2
COMMENTS
Previous name was: Triangle of numbers related to triangle A049223; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
T(n, m) = S2p(-4; 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) = A008546(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.
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 starts:
1;
4, 1;
36, 12, 1;
504, 192, 24, 1;
9576, 3960, 600, 40, 1;
229824, 100656, 17160, 1440, 60, 1;
6664896, 3048192, 563976, 54600, 2940, 84, 1;
226606464, 107255232, 21095424, 2256576, 142800, 5376, 112, 1;
MATHEMATICA
(* First program *)
T[n_, m_] /; n>=m>=1:= T[n, m]= (5*(n-1)-m)*T[n-1, m] + T[n-1, m-1]; T[n_, m_] /; n<m=0; T[_, 0]=0; T[1, 1]=1;
Table[T[n, m], {n, 10}, {m, n}]//Flatten (* Jean-François Alcover, Jun 20 2018 *)
(* Second program *)
rows = 10;
b[n_, m_]:= BellY[n, m, Table[k! Binomial[4, k], {k, 0, rows}]];
T= Table[b[n, m], {n, rows}, {m, rows}]//Inverse//Abs;
A011801= Table[T[[n, m]], {n, rows}, {m, n}]//Flatten (* Jean-François Alcover, Jun 22 2018 *)
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(4, n), 8) # Peter Luschny, Jan 16 2016
(Magma)
function T(n, k) // T = A011801
if k eq 0 then return 0;
elif k eq n then return 1;
else return (5*(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