A323434 Number of ways to split a strict integer partition of n into consecutive subsequences of equal length.

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 20, 24, 31, 38, 48, 59, 72, 86, 106, 125, 150, 180, 213, 250, 296, 347, 407, 477, 555, 645, 751, 869, 1003, 1161, 1334, 1534, 1763, 2018, 2306, 2637, 3002, 3418, 3886, 4409, 4994, 5659, 6390, 7214, 8135, 9160, 10300, 11580, 12990

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..7000

Formula

a(n) = Sum_y A000005(k), where the sum is over all strict integer partitions of n and k is the number of parts.

Example

The a(10) = 20 split partitions:
  [10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
.
  [9] [8] [7] [6] [4 3]
  [1] [2] [3] [4] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

Maple

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, numtheory[tau](t), b(n, i-1, t)+
         b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 15 2019

Mathematica

Table[Sum[Length[Divisors[Length[ptn]]], {ptn, Select[IntegerPartitions[n], UnsameQ@@#&]}], {n, 30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n>i(i+1)/2, 0,
     If[n == 0, DivisorSigma[0, t], b[n, i-1, t] +
     b[n-i, Min[n-i, i-1], t+1]]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 60] (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

Crossrefs

Cf. A000005, A000219, A101509, A117433, A316245, A319066, A319794, A323295, A323301, A323431, A323433.

Sequence in context: A050824 A323703 A333147 * A323431 A211516 A323430

Adjacent sequences: A323431 A323432 A323433 * A323435 A323436 A323437

Keyword

nonn

Author

Gus Wiseman, Jan 15 2019

Status

approved