A001452 Number of 5-line partitions of n.

1, 1, 3, 6, 13, 24, 47, 83, 152, 263, 457, 768, 1292, 2118, 3462, 5564, 8888, 14016, 21973, 34081, 52552, 80331, 122078, 184161, 276303, 411870, 610818, 900721, 1321848, 1929981, 2805338, 4058812, 5847966, 8390097, 11990531, 17069145, 24210571, 34215537, 48190451, 67644522

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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.

Crossrefs

A row of the array in A242641.

Sequences "number of r-line partitions": A000041 (r=1), A000990 (r=2), A000991 (r=3), A002799 (r=4), A001452 (r=5), A225196 (r=6), A225197 (r=7), A225198 (r=8), A225199 (r=9).

Programs

Magma

m:=50; R:=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

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 *)

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

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

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

Extensions

More terms from Sean A. Irvine, Jul 24 2012

a(0)=1 prepended by Joerg Arndt, May 01 2013