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A368116
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A(m, n) = lcm_{p in Partitions(n)} (Product_{r in p}(r + m)). Array read by ascending antidiagonals, for m, n >= 0.
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3
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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
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OFFSET
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0,5
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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, ...
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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.
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PROG
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(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)])
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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