OFFSET
3,5
LINKS
Alois P. Heinz, Table of n, a(n) for n = 3..1000
FORMULA
G.f.: Sum_{i>0} x^(2*i+3) / Product_{j=1..3+i} (1-x^j).
EXAMPLE
a(5) = 1: [4,1].
a(6) = 1: [4,1,1].
a(7) = 3: [4,1,1,1], [4,2,1], [5,2].
a(8) = 4: [4,1,1,1,1], [4,2,1,1], [4,3,1], [5,2,1].
a(9) = 8: [4,1,1,1,1,1], [4,2,1,1,1], [4,2,2,1], [4,3,1,1], [4,4,1], [5,2,1,1], [5,2,2], [6,3].
MAPLE
b:= proc(n, i) option remember;
`if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
end:
a:= n-> add(b(n-2*m-3, min(n-2*m-3, m+3)), m=1..(n-3)/2):
seq(a(n), n=3..60);
MATHEMATICA
Table[Count[IntegerPartitions[n], _?(MemberQ[#, #[[1]]-3]&)], {n, 3, 60}] (* Harvey P. Dale, Mar 01 2015 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
a[n_] := Sum[b[n - 2 m - 3, Min[n - 2 m - 3, m + 3]], {m, 1, (n - 3)/2}];
a /@ Range[3, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 20 2012
STATUS
approved