A327003 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.

1, 1, 1, 10, 1, 84, 280, 1, 220, 462, 9240, 15400, 1, 455, 5005, 50050, 210210, 1401400, 1401400, 1, 816, 18564, 185640, 24310, 4084080, 13613600, 2858856, 85765680, 285885600, 190590400, 1, 1330, 54264, 542640, 293930, 24690120, 82300400, 32332300, 135795660, 2715913200, 4526522000, 3802278480, 38022784800, 76045569600, 36212176000

Offset: 0

Author

Peter Luschny, Aug 14 2019

Keywords

• nonn
• tabf

Comments

The Hindenburg order refers to the partition generating algorithm of C. F. Hindenburg (1779). [Knuth 7.2.1.4H]

Examples

			The irregular triangle starts:
[0] [1]
[1] [1]
[2] [1, 10]
[3] [1, 84, 280]
[4] [1, 220, 462, 9240, 15400]
[5] [1, 455, 5005, 50050, 210210, 1401400, 1401400]
[6] [1, 816, 18564, 185640, 24310, 4084080, 13613600, 2858856, 85765680, 285885600, 190590400]
		

Crossrefs

Cf. A000012 (m=0, subdivided into rows of length A000041), A080575 (m=1), A257490 (m=2), this sequence (m=3), A327004 (m=4).

Cf. A000041 (length of rows), A291973 (sum of rows), A291451 (coarser subdivision).

Cf. A260876.

Programs

SageMath

def A327003row(n):
    shapes = ([3*x for x in p] for p in Partitions(n))
    return [SetPartitions(sum(s), s).cardinality() for s in shapes]
for n in (0..7): print(A327003row(n))

Formula

Row of lengths are in A000041.