A225197 Number of 7-line partitions of n (i.e., planar partitions of n with at most 7 lines).

1, 1, 3, 6, 13, 24, 48, 86, 159, 279, 492, 840, 1436, 2394, 3980, 6510, 10586, 17001, 27148, 42908, 67424, 105067, 162786, 250427, 383186, 582663, 881521, 1326319, 1986118, 2959376, 4390175, 6483255, 9534945, 13964910, 20374513, 29612085, 42883238, 61880879, 88993610, 127560266

Offset

0,3

Comments

Number of partitions of n where there are k sorts of parts k for k<=6 and seven 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

Vaclav Kotesovec, Graph - The asymptotic ratio (50000 terms, convergence is slow)

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

Formula

G.f.: 1/Product_{n>=1}(1-x^n)^min(n,7). - Joerg Arndt, Mar 15 2014

a(n) ~ 346032180025 * Pi^21 * sqrt(7) * exp(Pi*sqrt(14*n/3)) / (69984 * sqrt(3) * n^13). - Vaclav Kotesovec, Oct 28 2015

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      min(d, 7)*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, 7]*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:=7; 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 06 2018 *)

Prog

(PARI) x='x+O('x^66); r=7; Vec( prod(k=1, r-1, (1-x^k)^(r-k)) / eta(x)^r )
(Magma) m:=50; r:=7; 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 06 2018
(Sage)
m=50; r=7
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))^7
s.coefficients() # G. C. Greubel, Dec 06 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: A005405 A225196 A301597 * A225198 A225199 A000219

Adjacent sequences: A225194 A225195 A225196 * A225198 A225199 A225200

Keyword

nonn

Author

Joerg Arndt, May 01 2013

Status

approved