A244393 Number of partitions of n the largest part of which, call it m, appears once, m-1 appears at most twice, m-2 appears at most thrice, etc.

1, 1, 1, 2, 3, 4, 6, 9, 13, 17, 25, 33, 45, 61, 82, 106, 142, 183, 238, 306, 395, 499, 638, 804, 1014, 1268, 1586, 1967, 2447, 3018, 3721, 4566, 5598, 6827, 8328, 10108, 12257, 14812, 17884, 21508, 25856, 30980, 37076, 44261, 52776, 62768, 74578, 88407, 104681, 123703, 146018, 172019, 202445, 237830, 279087, 326991, 382706

Offset: 0

Author

David S. Newman, Jul 03 2014

Keywords

• nonn

Examples

			For n=6 the partitions counted are 6, 51, 42, 411, 321, 3111
The a(9) = 17 such partitions of 9 are:
01:  [ 3 2 2 1 1 ]
02:  [ 4 2 1 1 1 ]
03:  [ 4 2 2 1 ]
04:  [ 4 3 1 1 ]
05:  [ 4 3 2 ]
06:  [ 5 1 1 1 1 ]
07:  [ 5 2 1 1 ]
08:  [ 5 2 2 ]
09:  [ 5 3 1 ]
10:  [ 5 4 ]
11:  [ 6 1 1 1 ]
12:  [ 6 2 1 ]
13:  [ 6 3 ]
14:  [ 7 1 1 ]
15:  [ 7 2 ]
16:  [ 8 1 ]
17:  [ 9 ]
		

Links

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

Crossrefs

Cf. A244395.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1, `if`(t=1, 1, t+1))+add(
         b(n-i*j, i-1, t+1), j=1..min(t, n/i))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 29 2017

Mathematica

nend=20;
For[n=1,n<=nend,n++,
count[n]=0;
Ip=IntegerPartitions[n];
For[i=1,i<=Length[Ip],i++,
m=Max[Ip[[i]]];
condition=True;
Tip=Tally[Ip[[i]]];
For[j=1,j<=Length[Tip],j++,
condition=condition&&(Tip[[j]][[2]]<= m-Tip[[j]][[1]]+1)];
If[condition,count[n]++(*;Print[Ip[[i]]]*)]];
]
Table[count[i],{i,1,nend}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1, 0,
     b[n, i - 1, If[t == 1, 1, t + 1]] + Sum[
     b[n - i*j, i - 1, t + 1], {j, 1, Min[t, n/i]}]]];
a[n_] := b[n, n, 1];
a /@ Range[0, 60] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz *)

Extensions

More terms from Joerg Arndt, Jul 03 2014