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%I #13 Mar 08 2020 10:13:36
%S 1,0,1,0,2,2,0,6,2,3,0,24,12,3,4,0,120,20,12,4,5,0,720,112,42,16,5,6,
%T 0,5040,336,126,44,20,6,7,0,40320,2112,492,188,55,24,7,8,0,362880,
%U 11712,2802,640,215,66,28,8,9,0,3628800,92160,16938,3624,830,258,77,32,9,10
%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 largest part 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 Also T(n,k) = Sum_{p in P(n,k)} Cen(p) where Cen(p) is the size of the centralizer of any permutation of cycle type p.
%e Triangle starts:
%e [1]
%e [0, 1]
%e [0, 2, 2]
%e [0, 6, 2, 3]
%e [0, 24, 12, 3, 4]
%e [0, 120, 20, 12, 4, 5]
%e [0, 720, 112, 42, 16, 5, 6]
%e [0, 5040, 336, 126, 44, 20, 6, 7]
%e [0, 40320, 2112, 492, 188, 55, 24, 7, 8]
%o (Sage)
%o def A271708(n,k):
%o P = Partitions(n, max_part=k, inner=[k])
%o return sum([p.aut() for p in P])
%o for n in (0..9): print([A271708(n,k) for k in (0..n)])
%Y Cf. A110143 (row sums), A126074.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Apr 17 2016
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