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%I #10 Feb 29 2020 04:22:42
%S 1,1,1,10,1,84,280,1,220,462,9240,15400,1,455,5005,50050,210210,
%T 1401400,1401400,1,816,18564,185640,24310,4084080,13613600,2858856,
%U 85765680,285885600,190590400,1,1330,54264,542640,293930,24690120,82300400,32332300,135795660,2715913200,4526522000,3802278480,38022784800,76045569600,36212176000
%N Irregular triangle read by rows in which the n-th row lists multinomials for partitions of 3n which have only parts which are multiples of 3, in Hindenburg order.
%C The Hindenburg order refers to the partition generating algorithm of C. F. Hindenburg (1779). [Knuth 7.2.1.4H]
%F Row of lengths are in A000041.
%e The irregular triangle starts:
%e [0] [1]
%e [1] [1]
%e [2] [1, 10]
%e [3] [1, 84, 280]
%e [4] [1, 220, 462, 9240, 15400]
%e [5] [1, 455, 5005, 50050, 210210, 1401400, 1401400]
%e [6] [1, 816, 18564, 185640, 24310, 4084080, 13613600, 2858856, 85765680, 285885600, 190590400]
%o (SageMath)
%o def A327003row(n):
%o shapes = ([3*x for x in p] for p in Partitions(n))
%o return [SetPartitions(sum(s), s).cardinality() for s in shapes]
%o for n in (0..7): print(A327003row(n))
%Y Cf. A000012 (m=0, subdivided into rows of length A000041), A080575 (m=1), A257490 (m=2), this sequence (m=3), A327004 (m=4).
%Y Cf. A000041 (length of rows), A291973 (sum of rows), A291451 (coarser subdivision).
%Y Cf. A260876.
%K nonn,tabf
%O 0,4
%A _Peter Luschny_, Aug 14 2019