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A362635
Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i has size at least i.
5
1, 1, 1, 2, 5, 12, 31, 97, 351, 1318, 4963, 19391, 82531, 386704, 1926907, 9811733, 50252175, 261462430, 1415025895, 8118274255, 49355434511, 312266428040, 2012834117143, 13055850467371, 85215848844559, 565353777291346, 3866868949795579, 27548261709035105
OFFSET
0,4
LINKS
EXAMPLE
a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 123, 1|23.
a(4) = 5: 1234, 12|34, 13|24, 14|23, 1|234.
a(5) = 12: 12345, 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
a(6) = 31: 123456, 1234|56, 1235|46, 1236|45, 123|456, 1245|36, 1246|35, 124|356, 1256|34, 125|346, 126|345, 12|3456, 1345|26, 1346|25, 134|256, 1356|24, 135|246, 136|245, 13|2456, 1456|23, 145|236, 146|235, 14|2356, 156|234, 15|2346, 16|2345, 1|23456, 1|23|456, 1|24|356, 1|25|346, 1|26|345.
MAPLE
b:= proc(n, t) option remember; `if`(n=0, 1, add(
b(n-j, t+1)*binomial(n-1, j-1), j=t..n))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 28 2023
STATUS
approved