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

%I A304332 #30 May 13 2018 10:22:17
%S A304332 1,1,2,2,4,5,8,9,14,17,24,29,40,49,64,77,101,122,156,187,235,281,349,
%T A304332 416,514,608,742,877,1062,1252,1502,1766,2108,2467,2928,3419,4039,
%U A304332 4701,5524,6411,7505,8688,10130,11695,13587,15648,18118,20819,24034,27555,31712
%N A304332 Expansion of Product_{k>0} (1 + Sum_{m>0} x^(k*m!)).
%C A304332 Also the number of partitions of n in which each part occurs a factorial number of times.
%H A304332 Alois P. Heinz, <a href="/A304332/b304332.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from Seiichi Manyama)
%e A304332 n | Partitions of n in which each part occurs a factorial number of times
%e A304332 --+----------------------------------------------------------------------
%e A304332 1 | 1;
%e A304332 2 | 2 = 1+1;
%e A304332 3 | 3 = 2+1;
%e A304332 4 | 4 = 3+1 = 2+2 = 2+1+1;
%e A304332 5 | 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1;
%e A304332 6 | 6 = 5+1 = 4+2 = 4+1+1 = 3+2+1 = 3+3 = 2+2+1+1 = 1+1+1+1+1+1;
%e A304332 7 | 7 = 6+1 = 5+2 = 5+1+1 = 4+3 = 4+2+1 = 3+3+1 = 3+2+2 = 3+2+1+1;
%p A304332 b:= proc(n, i) option remember; local j; if n=0 then 1
%p A304332       elif i<1 then 0 else b(n, i-1); for j while
%p A304332         i*j!<=n do %+b(n-i*j!, i-1) od; % fi
%p A304332     end:
%p A304332 a:= n-> b(n$2):
%p A304332 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 11 2018
%Y A304332 Cf. A000041, A300446.
%K A304332 nonn
%O A304332 0,3
%A A304332 _Seiichi Manyama_, May 11 2018