A160974 Number of partitions of n where every part appears at least 4 times.

1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 4, 2, 4, 4, 7, 5, 8, 7, 13, 10, 13, 12, 21, 18, 22, 21, 34, 29, 40, 36, 55, 48, 63, 64, 88, 79, 100, 99, 139, 125, 160, 155, 207, 199, 241, 241, 314, 302, 369, 366, 466, 454, 550, 557, 690, 679, 807, 821, 1016, 1001, 1180, 1207, 1460, 1466, 1708

Offset: 0

Author

R. H. Hardin, Jun 01 2009

Keywords

• nonn

Examples

			a(12) = 4 because we have 3333, 2^6, 22221111, and 1^(12). - _Emeric Deutsch_, Jun 24 2009
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..967 from R. H. Hardin)

Crossrefs

Cf. A007690, A100405, A160975-A160990.

Programs

Maple

g := product(1+x^(4*j)/(1-x^j), j = 1..30): gser := series(g, x = 0, 85): seq(coeff(gser, x, n), n = 0..66); # Emeric Deutsch, Jun 24 2009
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=[0, $4..iquo(n, i)])))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Oct 02 2017

Mathematica

nmax = 100; CoefficientList[Series[Product[1 + x^(4*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2015; offset adapted by Georg Fischer, Sep 18 2020 *)

Formula

G.f.: Product_{j>=1} (1+x^(4*j)/(1-x^j)). - Emeric Deutsch, Jun 24 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(-4*x)) dx = -0.903005550655893892139378653023287247062261773608753265529... . - 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 Alois P. Heinz, Oct 02 2017