OFFSET
0,3
COMMENTS
Number of partitions of n where there are k sorts of parts k for k<=5 and six sorts of all other parts. - Joerg Arndt, Mar 15 2014
LINKS
Vincenzo Librandi and Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - The asymptotic ratio (50000 terms, convergence is slow)
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table II. - N. J. A. Sloane, May 21 2014
FORMULA
G.f.: 1/Product_{n>=1} (1-x^n)^min(n,6). - Joerg Arndt, Mar 15 2014
a(n) ~ 2160 * Pi^15 * exp(2*Pi*sqrt(n)) / n^(39/4). - Vaclav Kotesovec, Oct 28 2015
G.f.: (1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/( Prod_{j>=1} (1-x^j ) )^6. - G. C. Greubel, Dec 06 2018
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
min(d, 6)*d, d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..45); # Alois P. Heinz, Mar 15 2014
MATHEMATICA
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Min[d, 6]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Feb 18 2015, Alois P. Heinz *)
m:=50; CoefficientList[Series[(1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/( Product[(1-x^j), {j, 1, m}])^6, {x, 0, m}], x] (* G. C. Greubel, Dec 06 2018 *)
PROG
(PARI) x='x+O('x^66); r=6; Vec( prod(k=1, r-1, (1-x^k)^(r-k)) / eta(x)^r )
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/(&*[1-x^j: j in [1..2*m]] )^6 )); // G. C. Greubel, Dec 06 2018
(Sage)
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
s = (1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/prod(1-x^j for j in (1..60))^6
s.coefficients() # G. C. Greubel, Dec 06 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Joerg Arndt, May 01 2013
STATUS
approved