OFFSET
1,5
COMMENTS
a(n) = Sum_{k>=0} k*A240185(n,k).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
FORMULA
G.f.: g(x) = Sum_{i>=1} Sum_{j>=i+2} x^{i+j} * Sum_{k=i+1..j-1} (1-x^k) /((1-x^i)*(1-x^j)*Product_{k=i+1..j-1} (1-x^k)).
EXAMPLE
a(5) = 3 because for the partitions p = [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1] the sets of numbers between min(p) and max(p) that are not parts of the partition p are: empty, {2,3}, empty, {2}, empty, empty, empty.
MAPLE
g := add(add(x^(i+j)*add(1-x^k, k = i+1 .. j-1)/((1-x^i)*(1-x^j)*mul(1-x^k, k = i+1 .. j-1)), j = i+2 .. 80), i = 1 .. 80): gser := series(g, x = 0, 55): seq(coeff(gser, x, n), n = 1 .. 50);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(irem(n, i)=0,
`if`(t, [1, 0], [0$2]), [0$2])+ `if`(i>1, (p-> p+
`if`(t, [0, p[1]], 0))(b(n, i-1, t))+add(
b(n-i*j, i-1, true), j=1..(n-1)/i), 0)
end:
a:= n-> b(n$2, false)[2]:
seq(a(n), n=1..50); # Alois P. Heinz, Dec 30 2015
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[Mod[n, i] == 0, If[t, {1, 0}, {0, 0}], {0, 0}] + If[i > 1, Function[p, p + If[t, {0, p[[1]]}, 0]][b[n, i - 1, t]] + Sum[b[n - i*j, i - 1, True], {j, 1, (n - 1)/i}], 0]; a[n_] := b[n, n, False][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 30 2015
STATUS
approved