A335941 Number of partitions of n such that the set s of parts and multiplicities satisfies s = {1..max(s)}.
1, 1, 2, 1, 1, 4, 2, 5, 5, 9, 8, 15, 11, 14, 22, 28, 30, 36, 37, 53, 60, 80, 83, 104, 114, 148, 157, 201, 218, 283, 284, 362, 400, 455, 518, 624, 697, 807, 907, 1036, 1181, 1368, 1531, 1727, 1990, 2197, 2563, 2849, 3182, 3568, 4095, 4548, 5143, 5720, 6420
Offset: 0
Keywords
Examples
a(0) = 1: the empty partition. a(1) = 1: 1. a(2) = 2: 11, 2. a(3) = 1: 21. a(4) = 1: 211. a(5) = 4: 2111, 221, 311, 32. a(6) = 2: 2211, 321. a(7) = 5: 22111, 2221, 3211, 322, 331. a(8) = 5: 22211, 32111, 3221, 3311, 332. a(9) = 9: 222111, 321111, 32211, 3222, 33111, 3321, 42111, 4311, 432. a(10) = 8: 2221111, 322111, 32221, 331111, 33211, 4222, 4321, 433.
Links
- Chai Wah Wu, Table of n, a(n) for n = 0..158 (n = 0..120 from Alois P. Heinz)
Programs
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Maple
b:= proc(n,i,s) option remember; `if`(n=0, `if`(s={$0..max(s)}, 1, 0), `if`(i<1, 0, add( b(n-i*j, i-1, {s[], j, `if`(j=0, 0, i)}), j=0..n/i))) end: a:= n-> b(n, floor((sqrt(1+8*(n+1))-1)/2), {0}): seq(a(n), n=0..55); -
Mathematica
b[n_, i_, s_] := b[n, i, s] = If[n == 0, If[s == Range[0, Max[s]], 1, 0], If[i < 1, 0, Sum[ b[n-i*j, i-1, Union@Flatten@{s, j, If[j == 0, 0, i]}], {j, 0, n/i}]]]; a[n_] := b[n, Floor[(Sqrt[1 + 8*(n + 1)] - 1)/2], {0}]; Table[a[n], {n, 0, 55}] (* Jean-François Alcover, May 30 2022, after Alois P. Heinz *)