OFFSET
0,3
COMMENTS
Planar partitions into at most five rows. - Joerg Arndt, May 01 2013
Number of partitions of n where there are k sorts of parts k for k<=4 and 5 sorts all other parts. - Joerg Arndt, Mar 15 2014
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (first 1000 terms from Alois P. Heinz)
M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273.
Vaclav Kotesovec, Graph - The asymptotic ratio (50000 terms)
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.
FORMULA
G.f.: 1 / Product_{k>=1} (1-x^k)^min(k,5). - Sean A. Irvine, Jul 24 2012
a(n) ~ 15625 * Pi^10 * sqrt(5) * exp(Pi*sqrt(10*n/3)) / (2592 * sqrt(3) * n^7). - Vaclav Kotesovec, Oct 28 2015
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
min(d, 5)*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, 5]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
nmax = 40; CoefficientList[Series[(1-x)^4 * (1-x^2)^3 * (1-x^3)^2 * (1-x^4) * Product[1/(1-x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 28 2015 *)
PROG
(PARI) x='x+O('x^66); r=5; Vec( prod(k=1, r-1, (1-x^k)^(r-k)) / eta(x)^r ) \\ Joerg Arndt, May 01 2013
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)/(&*[1-x^j: j in [1..2*m]])^5 )); // G. C. Greubel, Dec 06 2018
(Sage)
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
s = (1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)/prod(1-x^j for j in (1..60))^5
list(s) # G. C. Greubel, Dec 06 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, Jul 24 2012
a(0)=1 prepended by Joerg Arndt, May 01 2013
STATUS
approved