reviewed

approved

proposed

reviewed

editing

proposed

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 *)

approved

editing

seq(seq(A111579(r, c), c=0..r), r=0..10) ; # _R. J. Mathar (mathar(AT)strw.leidenuniv.nl), _, Oct 30 2009

Edited by _R. J. Mathar (mathar(AT)strw.leidenuniv.nl), _, Oct 30 2009

_Gary W. Adamson (qntmpkt(AT)yahoo.com), _, Aug 07 2005

Generalized Bell number triangle, 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.

Generalized Triangles of generalized Stirling number numbers of the second kind triangles may be defined by the generating operation recurrences T(n,k) = T(n-1,k-1) + Q*T(n-1,k) where Q denotes an arithmetic sequence initialized by T(0,0)=T(1,0)=T(1,1,)=1... Q=1 generates Pascal's triangle); (1,2,3...Stirling number of the second kind triangle); (1,3,5...A039755 an analogue of the Stirling number of the second kind triangle); etc... 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).

Columns are row sums of generalized Stirling number of the second kind triangles.

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).

Column 2 (1, 2, 5, 15, 52, 203...are Bell numbers deleting the first 1); which are row sums of the Stirling number of the second kind triangle A008277.

Column 3 (1, 2, 6, 24, 116...) = row sums of A039755, a Stirling number of the second kind analogue.

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 (mathar(AT)strw.leidenuniv.nl), Oct 30 2009

nonn,tabl,uned,new

Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 30 2009

Generalized Bell number triangle, read by rows.

nonn,tabl,uned,new

Generalized Stirling number of the second kind triangles may be defined by the generating operation T(n,k) = T(n-1,k-1) + Q*T(n-1,k) where Q denotes an arithmetic sequence (1,1,1...Pascal's Triangletriangle); (1,2,3...Stirling number of the second kind triangle); (1,3,5...A039755 an analogue to of the Stirling number of the second kind triangle); etc...

nonn,tabl,uned,new

Generalized Bell number triangle, by rows.

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

0,5

Generalized Stirling number of the second kind triangles may be defined by the generating operation T(n,k) = T(n-1,k-1) + Q*T(n-1,k) where Q denotes an arithmetic sequence (1,1,1...Pascal's Triangle); (1,2,3...Stirling number of the second kind triangle); (1,3,5...A039755 an analogue to the Stirling number of the second kind triangle); etc...

Columns are row sums of generalized Stirling number of the second kind triangles.

Column 2 (1, 2, 5, 15, 52, 203...are Bell numbers deleting the first 1); which are row sums of the Stirling number of the second kind triangle A008277.

Column 3 (1, 2, 6, 24, 116...) = row sums of A039755, a Stirling number of the second kind analogue.

Cf. A008277, A000110, A039755, A004211, A111577, A111578.

nonn,tabl,uned

Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 07 2005

approved