OFFSET
0,5
COMMENTS
Triangles of generalized Stirling numbers of the second kind may be defined by recurrences T(n,k) = T(n-1,k-1) + Q*T(n-1,k) initialized by T(0,0)=T(1,0)=T(1,1)=1. Q=1 generates Pascal's triangle A007318,
Q=k+1 generates A008277, Q=2k+1 generates A039755, Q=3k+1 generates A111577, Q=4k+1 generates A111578, Q=5k+1 generates A166973.
(These definitions assume row and column enumeration 0<=n, 0<=k<=n.)
Each of these triangles characterized by Q=(c-1)*k+1 has row sums sum_{k=0..n} T(n,k), which define the column A(.,c).
FORMULA
MAPLE
T := proc(n, k, c) if k < 0 or k > n then 0 ; elif n <= 1 then 1; else procname(n-1, k-1, c)+((c-1)*k+1)*procname(n-1, k, c) ; fi; end:
A111579 := proc(r, c) local n; if c = 0 then 1 ; else n := r-c ; add( T(n, k, c), k=0..n) ; end if; end:
seq(seq(A111579(r, c), c=0..r), r=0..10) ; # R. J. Mathar, Oct 30 2009
MATHEMATICA
T[n_, k_, c_] := T[n, k, c] = If[k < 0 || k > n, 0, If[n <= 1, 1, T[n-1, k-1, c] + ((c-1)*k+1)*T[n-1, k, c]]];
A111579[r_, c_] := Module[{n}, If[c == 0, 1, n = r - c; Sum[T[n, k, c], {k, 0, n}]]];
Table[A111579[r, c], {r, 0, 10}, {c, 0, r}] // Flatten (* Jean-François Alcover, Aug 01 2023, after R. J. Mathar *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Aug 07 2005
EXTENSIONS
Edited by R. J. Mathar, Oct 30 2009
STATUS
approved