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A111579 Triangle A(r,c) read by rows, which contains the row sums of the triangle T(n,k)= T(n-1,k-1)+((c-1)*k+1)*T(n-1,k) in column c. 3
1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 5, 2, 1, 1, 16, 15, 6, 2, 1, 1, 32, 52, 24, 7, 2, 1, 1, 64, 203, 116, 35, 8, 2, 1, 1, 128, 877, 648, 214, 48, 9, 2, 1, 1, 256, 4140, 4088, 1523, 352, 63, 10, 2, 1, 1, 512, 21147, 28640, 12349, 3008, 536, 80, 11, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
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).
LINKS
FORMULA
A(r=n+c,c) = sum_{k=0..n} T(n,k,c), 0<=c<=r where T(n,k,c) = T(n-1,k-1,c) + ((c-1)*k+1)*T(n-1,k,c).
A(r,0) = 1.
A(r,1) = 2^(r-1).
A(r,2) = A000110(r-1).
A(r,3) = A007405(r-3).
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
Sequence in context: A176463 A098050 A278984 * A144374 A144018 A258709
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Aug 07 2005
EXTENSIONS
Edited by R. J. Mathar, Oct 30 2009
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)