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

0,4

Links

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

Example

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 ]

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 *)

Crossrefs

Cf. A244395.

Sequence in context: A240079 A240727 A123648 * A286929 A255525 A129632

Adjacent sequences: A244390 A244391 A244392 * A244394 A244395 A244396

Keyword

nonn

Author

David S. Newman, Jul 03 2014

Extensions

More terms from Joerg Arndt, Jul 03 2014

Status

approved