OFFSET
0,3
COMMENTS
Number of partitions of n where there are k sorts of parts k for k<=7 and eight 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
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
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms, convergence is very slow)
FORMULA
G.f.: 1/Product_{n>=1}(1-x^n)^min(n,8). - Joerg Arndt, Mar 15 2014
a(n) ~ 7696581394432000 * sqrt(2) * Pi^28 * exp(4*Pi*sqrt(n/3)) / (19683 * 3^(1/4) * n^(67/4)). - Vaclav Kotesovec, Oct 28 2015
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
min(d, 8)*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, 8]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
m:=50; r:=8; CoefficientList[Series[Product[(1-x^k)^(r-k), {k, 1, r-1}]/( Product[(1-x^j), {j, 1, m}])^r, {x, 0, m}], x] (* G. C. Greubel, Dec 10 2018 *)
PROG
(PARI) x='x+O('x^66); r=8; Vec( prod(k=1, r-1, (1-x^k)^(r-k)) / eta(x)^r )
(Magma) m:=50; r:=8; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1-x^k)^(r-k): k in [1..r-1]])/(&*[1-x^j: j in [1..2*m]] )^r )); // G. C. Greubel, Dec 10 2018
(Sage)
m=50; r=8
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(m)
s = prod((1-x^k)^(r-k) for k in (1..r-1))/prod(1-x^j for j in (1..m+2))^r
s.coefficients() # G. C. Greubel, Dec 10 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Joerg Arndt, May 01 2013
STATUS
approved