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Number of 7-line partitions of n (i.e., planar partitions of n with at most 7 lines).
10

%I #47 Sep 08 2022 08:46:04

%S 1,1,3,6,13,24,48,86,159,279,492,840,1436,2394,3980,6510,10586,17001,

%T 27148,42908,67424,105067,162786,250427,383186,582663,881521,1326319,

%U 1986118,2959376,4390175,6483255,9534945,13964910,20374513,29612085,42883238,61880879,88993610,127560266

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

%C 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

%H Vincenzo Librandi and Joerg Arndt and Alois P. Heinz, <a href="/A225197/b225197.txt">Table of n, a(n) for n = 0..1000</a>

%H Vaclav Kotesovec, <a href="/A225197/a225197.jpg">Graph - The asymptotic ratio</a> (50000 terms, convergence is slow)

%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,7). - _Joerg Arndt_, Mar 15 2014

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

%p with(numtheory):

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

%p min(d, 7)*d, d=divisors(j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..45); # _Alois P. Heinz_, Mar 15 2014

%t 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_ *)

%t 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 *)

%o (PARI) x='x+O('x^66); r=7; Vec( prod(k=1,r-1, (1-x^k)^(r-k)) / eta(x)^r )

%o (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

%o (Sage)

%o m=50; r=7

%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))^7

%o s.coefficients() # _G. C. Greubel_, Dec 06 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