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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

Table of n, a(n) for n=0..65.

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

CROSSREFS

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

Sequence in context: A176463 A098050 A278984 * A144374 A144018 A258709

Adjacent sequences:  A111576 A111577 A111578 * A111580 A111581 A111582

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 January 25 09:48 EST 2021. Contains 340416 sequences. (Running on oeis4.)