A325874 - OEIS

A325874

Number of integer partitions of n whose differences of all degrees > 1 are nonzero.

1, 1, 2, 2, 4, 5, 6, 8, 12, 13, 19, 24, 26, 33, 45, 52, 66, 78, 92, 113, 129, 160, 192, 231, 268, 305, 361, 436, 501, 591, 665, 783, 897, 1071, 1228, 1361, 1593, 1834, 2101, 2452, 2685, 3129, 3526, 4067, 4568, 5189, 5868, 6655, 7565, 8468, 9400

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OFFSET

0,3

COMMENTS

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. If m is the length of the sequence, its differences of all degrees are the union of the zeroth through m-th differences.

The case for all degrees including 1 is A325852.

LINKS

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..220

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts

EXAMPLE

The a(1) = 1 through a(9) = 13 partitions:

(1) (2) (3) (4) (5) (6) (7) (8) (9)

(11) (21) (22) (32) (33) (43) (44) (54)

(31) (41) (42) (52) (53) (63)

(211) (221) (51) (61) (62) (72)

(311) (411) (322) (71) (81)

(2211) (331) (332) (441)