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A320509 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part. 2
1, 1, 2, 3, 3, 4, 6, 4, 6, 8, 7, 8, 11, 7, 12, 14, 10, 13, 19, 12, 18, 21, 16, 19, 27, 19, 25, 30, 25, 30, 37, 25, 35, 40, 35, 42, 49, 35, 49, 56, 46, 54, 66, 50, 65, 72, 60, 70, 83, 68, 84, 90, 80, 94, 110, 86, 107, 116, 98, 119, 137, 111, 134, 146, 130, 148, 165, 141, 169 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Partitions are usually written with parts in descending order, but the conditions are easier to check visually if written in ascending order.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..300

EXAMPLE

There are a(11) = 8 such partitions of 11:

01: [11]

02: [4, 7]

03: [5, 6]

04: [2, 4, 5]

05: [3, 4, 4]

06: [2, 3, 3, 3]

07: [1, 2, 2, 2, 2, 2]

08: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

There are a(12) = 11 such partitions of 12:

01: [12]

02: [4, 8]

03: [5, 7]

04: [6, 6]

05: [2, 4, 6]

06: [3, 4, 5]

07: [4, 4, 4]

08: [3, 3, 3, 3]

09: [1, 2, 3, 3, 3]

10: [2, 2, 2, 2, 2, 2]

11: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

PROG

(Ruby)

def partition(n, min, max)

  return [[]] if n == 0

  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}

end

def f(n)

  return 1 if n == 0

  cnt = 0

  partition(n, 1, n).each{|ary|

    ary << 0

    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}

    cnt += 1 if ary0.sort == ary0

  }

  cnt

end

def A320509(n)

  (0..n).map{|i| f(i)}

end

p A320509(50)

CROSSREFS

Cf. A240026, A240027, A320466, A320470, A320510.

Cf. A320387 (distinct parts, nonincreasing, and first difference <= first part).

Sequence in context: A200763 A203291 A220053 * A159999 A003977 A003971

Adjacent sequences:  A320506 A320507 A320508 * A320510 A320511 A320512

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 14 2018

STATUS

approved

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Last modified March 23 16:52 EDT 2019. Contains 321432 sequences. (Running on oeis4.)