A368116 A(m, n) = lcm_{p in Partitions(n)} (Product_{r in p}(r + m)). Array read by ascending antidiagonals, for m, n >= 0.

1, 1, 1, 1, 2, 2, 1, 3, 12, 6, 1, 4, 36, 24, 12, 1, 5, 80, 540, 720, 60, 1, 6, 150, 960, 6480, 1440, 360, 1, 7, 252, 5250, 134400, 136080, 60480, 2520, 1, 8, 392, 1512, 315000, 537600, 8164800, 120960, 5040, 1, 9, 576, 24696, 63504, 1575000, 32256000, 24494400, 3628800, 15120

Offset: 0

Author

Peter Luschny, Dec 12 2023

Keywords

• nonn
• tabl

Comments

We say q is a 'm-shifted partition of n' if there is a partition p of n, p = (t1, t2, ..., tk) and q = (t1 + m, t2 + m, ..., tk + m), where m is a nonnegative integer. q is a partition of n + k*m.

Let P(n) denote the partitions of n and P_{m}(n) the m-shifted partitions of n. The product of a partition is the product of its parts, Prod(p) = p1*p2*...*pk if p = (p1, p2, ..., pk). Using this terminology, the definition can be written as A(m, n) = lcm_{p in P_{m}(n)} Prod(p).

With m = 0 the cumulative radical A048803 is computed, and with m = 1 the Hirzebruch numbers A091137.

Examples

			Array A(m, n) begins:
  [0] 1, 1,   2,     6,       12,        60,           360, ...  A048803
  [1] 1, 2,  12,    24,      720,      1440,         60480, ...  A091137
  [2] 1, 3,  36,   540,     6480,    136080,       8164800, ...  A368048
  [3] 1, 4,  80,   960,   134400,    537600,      32256000, ...
  [4] 1, 5, 150,  5250,   315000,   1575000,     330750000, ...
  [5] 1, 6, 252,  1512,    63504,   1905120,     880165440, ...
  [6] 1, 7, 392, 24696,  6914880, 532445760,  268352663040, ...
  [7] 1, 8, 576, 23040, 18247680, 145981440,  683193139200, ...
  [8] 1, 9, 810, 80190,  7217100, 844400700, 5851696851000, ...
.
Let m = 2 and n = 4. The partitions of 4 are [(4), (3,1), (2,2), (2,1,1), (1, 1, 1, 1)]. Thus A(2, 4) = lcm([6, 5*3, 4*4, 4*3*3, 3*3*3*3]) = 6480.
		

Crossrefs

Cf. A048803 (m=0), A091137 (m=1), A368048 (m=2).

Columns include: A000027, A011379.

Programs

SageMath

def A(m, n): return lcm(product(r + m for r in p) for p in Partitions(n))
for m in range(9): print([A(m, n) for n in range(7)])