A277101 Sum over all partitions of n of the number of distinct parts i of multiplicity i - 1.

0, 0, 1, 1, 1, 2, 4, 5, 8, 10, 15, 20, 29, 37, 52, 67, 89, 115, 152, 192, 251, 316, 405, 508, 644, 799, 1006, 1243, 1546, 1901, 2351, 2871, 3527, 4289, 5232, 6336, 7688, 9264, 11189, 13430, 16137, 19299, 23097, 27514, 32799, 38944, 46246, 54738, 64782, 76430, 90171

Offset: 0

Author

Emeric Deutsch, Oct 10 2016

Keywords

• nonn

Examples

			a(6) = 4 because in the partitions [1,1,1,1,1,1], [1,1,1,1,2'], [1,1,2,2], [2,2,2], [1,1,1,3], [1,2',3], [3',3], [1,1,4], [2',4], [1,5], [6] of 6 only the marked parts satisfy the requirement.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Crossrefs

Cf. A276427, A276428, A276429, A276433, A276434, A277099, A277100, A277102.

Programs

Maple

g := (sum(x^(i*(i+1))*(1-x^(i+1)), i = 1 .. 200))/(product(1-x^i, i = 1 .. 200)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 0 .. 50);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
     `if`(i<1, 0, add((p-> p+`if`(i-1<>j, 0,
      [0, p[1]]))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 10 2016

Mathematica

max = 60; s = Sum[x^(i*(i+1))*(1-x^(1+i)), {i, 1, max}]/QPochhammer[x] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 08 2016 *)

Formula

a(n) = Sum(k*A277100(n,k), k>=0).

G.f.: g(x) = Sum_(i>=1)(x^(i(i+1))(1-x^(i+1)))/Product_(i>=1)(1-x^i).