A160975 Number of partitions of n where every part appears at least 5 times.

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 3, 3, 5, 4, 7, 7, 7, 8, 11, 12, 12, 14, 15, 16, 23, 20, 24, 26, 29, 36, 40, 40, 46, 50, 63, 63, 76, 76, 87, 103, 108, 117, 135, 140, 167, 173, 191, 205, 235, 257, 278, 300, 327, 354, 413, 424, 469, 511, 555, 616, 673, 711, 783, 849, 947

Offset: 0

Author

R. H. Hardin, Jun 01 2009

Keywords

• nonn

Examples

			a(15) = 3 because we have 33333, 2222211111, and 1^(15). - _Emeric Deutsch_, Jun 28 2009
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..5000 (terms 1..1000 from R. H. Hardin)

Crossrefs

Cf. A007690, A100405, A160974, A160976-A160990.

Programs

Maple

g := product(1+x^(5*j)/(1-x^j), j = 1..20): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0..75); # Emeric Deutsch, Jun 28 2009
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(b(n-i*j, i-1), j=5..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, Feb 06 2024

Mathematica

nmax = 100; Rest[CoefficientList[Series[Product[1 + x^(5*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 28 2015 *)

Formula

G.f.: Product_{j>=1} (1+x^(5*j)/(1-x^j)). - Emeric Deutsch, Jun 28 2009

a(n) ~ sqrt(Pi^2 + 6*c) * exp(sqrt((2*Pi^2/3 + 4*c)*n)) / (4*sqrt(3)*Pi*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-5*x)) dx = -0.990807844177842472956484606320623872921836802804155824925... . - Vaclav Kotesovec, Jan 05 2016

Extensions

Initial terms changed to match b-file. - N. J. A. Sloane, Aug 31 2009

Maple program fixed by Vaclav Kotesovec, Nov 28 2015

a(0)=1 prepended by Seiichi Manyama, Feb 06 2024