A277102 Number of partitions of n containing no part i of multiplicity i-1.

1, 1, 1, 2, 4, 5, 7, 10, 15, 21, 28, 37, 50, 67, 88, 115, 150, 193, 248, 317, 402, 508, 640, 802, 1002, 1248, 1545, 1908, 2351, 2887, 3532, 4313, 5251, 6377, 7724, 9334, 11254, 13541, 16253, 19473, 23286, 27791, 33100, 39362, 46723, 55370, 65504, 77377, 91257, 107477, 126380

Offset: 0

Author

Emeric Deutsch, Oct 10 2016

Keywords

• nonn

Examples

			a(4) = 4 because we have [1,1,1,1], [1,3], [2,2], and [4]; the partition [1,1,2] does not qualify.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Alois P. Heinz)

Crossrefs

Cf. A276427, A276428, A276429, A276433, A276434, A277099, A277100, A277101.

Programs

Maple

g := (product(1/(1-x^(i+1))-x^(i*(i+1)), i = 1 .. 100))/(1-x): 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, `if`(i<1, 0,
      add(`if`(i-1=j, 0, b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 10 2016

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[If[i-1 == j, 0, b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 11 2016 after Alois P. Heinz *)

Formula

a(n) = A277100(n,0).

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