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A325874 Number of integer partitions of n whose differences of all degrees > 1 are nonzero. 10
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 (list; graph; refs; listen; history; text; internal format)
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)

                                         (421)  (422)   (522)

                                         (511)  (431)   (621)

                                                (521)   (711)

                                                (611)   (4221)

                                                (3221)  (4311)

                                                (3311)  (5211)

                                                        (32211)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], !MemberQ[Union@@Table[Differences[#, i], {i, 2, Length[#]}], 0]&]], {n, 0, 30}]

CROSSREFS

Cf. A049988, A238423, A325325, A325468, A325545, A325849, A325850, A325851, A325852, A325875, A325876.

Sequence in context: A266745 A062436 A238571 * A293957 A238687 A238433

Adjacent sequences:  A325871 A325872 A325873 * A325875 A325876 A325877

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 02 2019

STATUS

approved

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Last modified February 28 20:48 EST 2021. Contains 341731 sequences. (Running on oeis4.)