A277099 Number of partitions of n containing no part i of multiplicity i+1.

1, 1, 1, 3, 4, 6, 8, 12, 18, 24, 32, 45, 59, 79, 104, 137, 177, 229, 295, 377, 477, 605, 761, 956, 1193, 1484, 1840, 2276, 2800, 3441, 4210, 5141, 6261, 7603, 9206, 11132, 13419, 16144, 19380, 23223, 27763, 33134, 39467, 46931, 55703, 66008, 78085, 92239, 108776, 128091, 150617

Offset: 0

Author

Emeric Deutsch, Sep 30 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

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

Crossrefs

Cf. A276427, A276428, A276429, A276433, A276434, A277100, A277101, A277102.

Programs

Maple

g:= product(1/(1-x^i)-x^(i*(i+1)), i = 1 .. 100): 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, Sep 30 2016

Mathematica

nmax = 50; CoefficientList[Series[Product[(1/(1-x^k) - x^(k*(k+1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 30 2016 *)

Formula

a(n) = A276433(n,0).

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