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

0,4

Comments

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

Links

Table of n, a(n) for n=0..44.

Formula

Row of lengths are in A000041.

Example

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]

Prog

(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))

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.

Sequence in context: A009227 A305996 A030526 * A206819 A178865 A347491

Adjacent sequences: A327000 A327001 A327002 * A327004 A327005 A327006

Keyword

nonn,tabf

Author

Peter Luschny, Aug 14 2019

Status

approved