A225199 Number of 9-line partitions of n (i.e., planar partitions of n with at most 9 lines).

1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 499, 856, 1471, 2466, 4124, 6788, 11110, 17965, 28890, 45995, 72819, 114354, 178577, 276952, 427279, 655199, 999773, 1517388, 2292377, 3446462, 5159352, 7689517, 11414606, 16875813, 24856366, 36474188, 53334376, 77717219, 112874158, 163403202

Offset: 0

Author

Joerg Arndt, May 01 2013

Keywords

• nonn

Comments

Number of partitions of n where there are k sorts of parts k for k<=8 and nine sorts of all other parts. - Joerg Arndt, Mar 15 2014

In general, "number of r-line partitions" is asymptotic to (Product_{j=1..r-1} j!) * Pi^(r*(r-1)/2) * r^((r^2 + 1)/4) * exp(Pi*sqrt(2*n*r/3)) / (2^((r*(r+2)+5)/4) * 3^((r^2 + 1)/4) * n^((r^2 + 3)/4)). - Vaclav Kotesovec, Oct 28 2015

Links

• Vincenzo Librandi and Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020, p. 28.
• 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

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:=9; R:=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

Maple

b:= proc(n,i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(min(i, 9)+j-1, j)*
       b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 15 2014

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Min[d, 9]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
m:=50; r:=9; 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 *)

PARI

x='x+O('x^66); r=9; Vec( prod(k=1,r-1, (1-x^k)^(r-k)) / eta(x)^r )

Sage

m=50; r=9
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

Formula

G.f.: 1/Product_{n>=1}(1-x^n)^min(n,9). - Joerg Arndt, Mar 15 2014

a(n) ~ 2101805306799541875 * sqrt(3) * Pi^36 * exp(Pi*sqrt(6*n)) / (8*n^21). [The convergence is very slow, numerical verification needs more than 1000000 terms.] - Vaclav Kotesovec, Oct 28 2015