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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.

%I A111579 #10 Aug 01 2023 07:39:43
%S A111579 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,
%T A111579 203,116,35,8,2,1,1,128,877,648,214,48,9,2,1,1,256,4140,4088,1523,352,
%U A111579 63,10,2,1,1,512,21147,28640,12349,3008,536,80,11,2,1
%N 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.
%C A111579 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,
%C A111579 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.
%C A111579 (These definitions assume row and column enumeration 0<=n, 0<=k<=n.)
%C A111579 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).
%F A111579 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).
%F A111579 A(r,0) = 1.
%F A111579 A(r,1) = 2^(r-1).
%F A111579 A(r,2) = A000110(r-1).
%F A111579 A(r,3) = A007405(r-3).
%p A111579 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:
%p A111579 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:
%p A111579 seq(seq(A111579(r,c),c=0..r),r=0..10) ; # _R. J. Mathar_, Oct 30 2009
%t A111579 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]]];
%t A111579 A111579[r_, c_] := Module[{n}, If[c == 0, 1, n = r - c; Sum[T[n, k, c], {k, 0, n}]]];
%t A111579 Table[A111579[r, c], {r, 0, 10}, {c, 0, r}] // Flatten (* _Jean-François Alcover_, Aug 01 2023, after _R. J. Mathar_ *)
%Y A111579 Cf. A008277, A000110, A039755, A004211, A111577, A111578.
%K A111579 nonn,tabl
%O A111579 0,5
%A A111579 _Gary W. Adamson_, Aug 07 2005
%E A111579 Edited by _R. J. Mathar_, Oct 30 2009