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%I #10 Mar 13 2020 16:57:31
%S 1,0,1,0,2,2,0,3,2,6,0,4,11,4,24,0,5,10,14,12,120,0,6,31,62,34,48,720,
%T 0,7,28,60,84,120,240,5040,0,8,66,102,490,228,552,1440,40320,0,9,60,
%U 299,292,708,912,3120,10080,362880,0,10,120,282,722,4396,2136,4752,20880,80640,3628800
%N Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with length k and Aut(p) = 1^j[1]*j[1]!*...*n^j[n]*j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.
%C S(n,k) = Sum_{p in P(n,k)} n!/Aut(p) are the Stirling cycle numbers A132393.
%e Triangle starts:
%e [1]
%e [0, 1]
%e [0, 2, 2]
%e [0, 3, 2, 6]
%e [0, 4, 11, 4, 24]
%e [0, 5, 10, 14, 12, 120]
%e [0, 6, 31, 62, 34, 48, 720]
%e [0, 7, 28, 60, 84, 120, 240, 5040]
%o (Sage)
%o def A271707(n,k):
%o P = Partitions(n, length=k)
%o return sum(p.aut() for p in P)
%o for n in (0..10): print([A271707(n,k) for k in (0..n)])
%Y Cf. A110143 (row sums), A132393, A271708.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Apr 17 2016
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