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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

%I

%S 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 203,116,35,8,2,1,1,128,877,648,214,48,9,2,1,1,256,4140,4088,1523,352,

%U 63,10,2,1,1,512,21147,28640,12349,3008,536,80,11,2,1

%N 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 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 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 (These definitions assume row and column enumeration 0<=n, 0<=k<=n.)

%C 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 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 A(r,0) = 1.

%F A(r,1) = 2^(r-1).

%F A(r,2) = A000110(r-1).

%F A(r,3) = A007405(r-3).

%p 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 := 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 seq(seq(A111579(r,c),c=0..r),r=0..10) ; # _R. J. Mathar_, Oct 30 2009

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

%K nonn,tabl

%O 0,5

%A _Gary W. Adamson_, Aug 07 2005

%E Edited by _R. J. Mathar_, Oct 30 2009

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Last modified March 2 12:26 EST 2021. Contains 341750 sequences. (Running on oeis4.)