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A325874 Number of integer partitions of n whose differences of all degrees > 1 are nonzero. 7
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

Table of n, a(n) for n=0..50.

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 25 04:58 EST 2020. Contains 332217 sequences. (Running on oeis4.)