A225198 Number of 8-line partitions of n (i.e., planar partitions of n with at most 8 lines).

1, 1, 3, 6, 13, 24, 48, 86, 160, 281, 497, 851, 1460, 2442, 4076, 6692, 10928, 17623, 28266, 44873, 70842, 110910, 172674, 266942, 410512, 627387, 954113, 1443063, 2172456, 3254446, 4854236, 7208018, 10659872, 15700111, 23035956, 33671399, 49042600, 71179250, 102963936, 148452294

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

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).

Sequence in context: A225196 A301597 A225197 * A225199 A000219 A356941

Adjacent sequences: A225195 A225196 A225197 * A225199 A225200 A225201

Keyword

nonn

Author

Joerg Arndt, May 01 2013

Status

approved