OFFSET
0,7
COMMENTS
Or number of partitions of n where part i has multiplicity < i and all multiplicities are distinct and different from all parts.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
EXAMPLE
a(0) = 1: the empty sum.
a(6) = 2: 1*6 = 2*3.
a(8) = 2: 1*8 = 2*4.
a(10) = 3: 1*10 = 2*5 = 1*4+2*3.
a(11) = 3: 1*11 = 1*5+2*3 = 2*4+1*3.
a(12) = 4: 1*12 = 2*6 = 1*6+2*3 = 3*4.
a(13) = 4: 1*13 = 1*7+2*3 = 2*5+1*3 = 1*5+2*4.
a(14) = 6: 1*14 = 1*8+2*3 = 2*7 = 1*6+2*4 = 2*5+1*4 = 3*4+1*2.
a(15) = 5: 1*15 = 1*9+2*3 = 1*7+2*4 = 2*6+1*3 = 3*5.
a(25) = 14: 1*25 = 1*19+2*3 = 1*17+2*4 = 1*15+2*5 = 1*13+2*6 = 1*13+3*4 = 2*11+1*3 = 1*11+2*7 = 2*10+1*5 = 1*10+3*5 = 2*9+1*7 = 1*9+2*8 = 3*7+1*4 = 1*7+3*6.
MAPLE
h:= proc(n) option remember;
(((2*n+3)*n-2)*n-`if`(n::odd, 3, 0))/12
end:
g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0,
b(n, i, select(x-> x<=i, s)))):
b:= proc(n, i, s) option remember; g(n, i-1, s)+
`if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j,
min(n-i*j, i-1), s union {j})), j=1..min(i-1, n/i)))
end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..100);
MATHEMATICA
h[n_] := h[n] = (((2*n + 3)*n - 2)*n - If[OddQ[n], 3, 0])/12;
g[n_, i_, s_] := If[n==0, 1, If[n>h[i], 0, b[n, i, Select[s, # <= i&]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] + If[MemberQ[s, i], 0, Sum[If[MemberQ[s, j], 0, g[n - i*j, Min[n - i*j, i - 1], s ~Union~ {j}]], {j, 1, Min[i - 1, n/i]}]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 01 2018, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 09 2017
STATUS
approved