A100882 Number of partitions of n in which the sequence of frequencies of the summands is nonincreasing.

1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 18, 21, 29, 29, 40, 47, 56, 62, 83, 86, 111, 124, 146, 166, 207, 217, 267, 300, 352, 389, 471, 505, 604, 668, 772, 860, 1015, 1085, 1279, 1419, 1622, 1780, 2072, 2242, 2595, 2858, 3231, 3563, 4092, 4421, 5057, 5557, 6250

Offset: 0

Author

David S. Newman, Nov 21 2004

Keywords

• nonn

Examples

			a(4) = 4 because in each of the partitions 4, 3+1, 2+2, 1+1+1+1, the frequencies of the summands is nonincreasing as the summands decrease. The partition 2+1+1 is not counted because 2 is used once, but 1 is used twice.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A100881, A100883, A100884.

Programs

Maple

b:= proc(n,i,t) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=1 then `if`(n<=t, 1, 0)
    elif i=0 then 0
    else      b(n, i-1, t)
         +add(b(n-i*j, i-1, j), j=1..min(t, floor(n/i)))
      fi
    end:
a:= n-> b(n, n, n):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 21 2011

Mathematica

b[n_, i_, t_] := b[n, i, t] = Which[n<0, 0, n == 0, 1, i == 1, If[n <= t, 1, 0], i == 0, 0, True, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, 1, Min[t, Floor[n/i]]}]]; a[n_] := b[n, n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Feb 21 2011