A225199 - OEIS

%I #44 Jun 17 2024 03:50:16

%S 1,1,3,6,13,24,48,86,160,282,499,856,1471,2466,4124,6788,11110,17965,

%T 28890,45995,72819,114354,178577,276952,427279,655199,999773,1517388,

%U 2292377,3446462,5159352,7689517,11414606,16875813,24856366,36474188,53334376,77717219,112874158,163403202

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

%C 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

%C 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

%H Vincenzo Librandi and Joerg Arndt and Alois P. Heinz, <a href="/A225199/b225199.txt">Table of n, a(n) for n = 0..1000</a>

%H Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, <a href="https://arxiv.org/abs/2004.08901">Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions</a>, arXiv:2004.08901 [math.CO], 2020, p. 28.

%H P. A. MacMahon, <a href="https://archive.org/stream/messengerofmathe52cambuoft#page/112/mode/2up">The connexion between the sum of the squares of the divisors and the number of partitions of a given number</a>, 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

%F G.f.: 1/Product_{n>=1}(1-x^n)^min(n,9). - _Joerg Arndt_, Mar 15 2014

%F 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

%p b:= proc(n,i) option remember; `if`(n=0, 1,

%p       `if`(i<1, 0, add(binomial(min(i, 9)+j-1, j)*

%p        b(n-i*j, i-1), j=0..n/i)))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Mar 15 2014

%t 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_ *)

%t 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 *)

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

%o (Magma) m:=50; r:=9; 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

%o (Sage)

%o m=50; r=9

%o R = PowerSeriesRing(ZZ, 'x')

%o x = R.gen().O(m)

%o s = (prod((1-x^k)^(r-k) for k in (1..r-1))/prod(1-x^j for j in (1..m+2))^r)

%o s.coefficients() # _G. C. Greubel_, Dec 10 2018

%Y A row of the array in A242641.

%Y 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).

%K nonn

%O 0,3

%A _Joerg Arndt_, May 01 2013