A265257 Number of odd singletons in all partitions of n (n>=0).

0, 1, 0, 2, 2, 5, 5, 11, 13, 23, 28, 45, 57, 86, 108, 156, 199, 276, 350, 475, 601, 798, 1005, 1312, 1646, 2120, 2643, 3365, 4178, 5264, 6500, 8122, 9981, 12375, 15136, 18638, 22697, 27779, 33679, 40993, 49504, 59947, 72109

Offset: 0

Author

Emeric Deutsch, Jan 01 2016

Keywords

• nonn

Examples

			a(6) = 5 because in [1,1,1,3], [1,2,3], [1,5] we have 1+2+2 odd singletons, while the other 8 partitions of 6 have no odd singletons.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A265255.

Programs

Maple

g := x*(1-x+x^2)/((1-x^4)*mul(1-x^j, j = 1 .. 80)): gser := series(g, x = 0, 55): seq(coeff(gser, x, m), m = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, add((p-> `if`(j=1 and i::odd, p+
      [0, p[1]], p))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..80);  # Alois P. Heinz, Jan 01 2016

Mathematica

nmax = 50; CoefficientList[Series[x*(1-x+x^2)/(1-x^4) * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 01 2016 *)

Formula

a(n) = Sum(k*A265255(n,k), k>=0).

G.f.: g(x) = x(1 - x + x^2)/((1-x^4)*Product_{j>=1}(1-x^j)).

From Vaclav Kotesovec, Jan 01 2016: (Start)

a(n) = 1/4 * A000070(n) - 3/4 * A087787(n) + 1/2 * A092295(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*Pi*sqrt(2*n)).

(End)