A304332 Expansion of Product_{k>0} (1 + Sum_{m>0} x^(k*m!)).

1, 1, 2, 2, 4, 5, 8, 9, 14, 17, 24, 29, 40, 49, 64, 77, 101, 122, 156, 187, 235, 281, 349, 416, 514, 608, 742, 877, 1062, 1252, 1502, 1766, 2108, 2467, 2928, 3419, 4039, 4701, 5524, 6411, 7505, 8688, 10130, 11695, 13587, 15648, 18118, 20819, 24034, 27555, 31712

Offset: 0

Author

Seiichi Manyama, May 11 2018

Keywords

• nonn

Comments

Also the number of partitions of n in which each part occurs a factorial number of times.

Examples

			n | Partitions of n in which each part occurs a factorial number of times
--+----------------------------------------------------------------------
1 | 1;
2 | 2 = 1+1;
3 | 3 = 2+1;
4 | 4 = 3+1 = 2+2 = 2+1+1;
5 | 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1;
6 | 6 = 5+1 = 4+2 = 4+1+1 = 3+2+1 = 3+3 = 2+2+1+1 = 1+1+1+1+1+1;
7 | 7 = 6+1 = 5+2 = 5+1+1 = 4+3 = 4+2+1 = 3+3+1 = 3+2+2 = 3+2+1+1;
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Seiichi Manyama)

Crossrefs

Cf. A000041, A300446.

Programs

Maple

b:= proc(n, i) option remember; local j; if n=0 then 1
      elif i<1 then 0 else b(n, i-1); for j while
        i*j!<=n do %+b(n-i*j!, i-1) od; % fi
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 11 2018