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

%I A225198 #36 Sep 08 2022 08:46:04
%S A225198 1,1,3,6,13,24,48,86,160,281,497,851,1460,2442,4076,6692,10928,17623,
%T A225198 28266,44873,70842,110910,172674,266942,410512,627387,954113,1443063,
%U A225198 2172456,3254446,4854236,7208018,10659872,15700111,23035956,33671399,49042600,71179250,102963936,148452294
%N A225198 Number of 8-line partitions of n (i.e., planar partitions of n with at most 8 lines).
%C A225198 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
%H A225198 Vincenzo Librandi and Joerg Arndt and Alois P. Heinz, <a href="/A225198/b225198.txt">Table of n, a(n) for n = 0..1000</a>
%H A225198 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
%H A225198 Vaclav Kotesovec, <a href="/A225198/a225198.jpg">Graph - The asymptotic ratio</a> (100000 terms, convergence is very slow)
%F A225198 G.f.: 1/Product_{n>=1}(1-x^n)^min(n,8). - _Joerg Arndt_, Mar 15 2014
%F A225198 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
%p A225198 with(numtheory):
%p A225198 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A225198       min(d, 8)*d, d=divisors(j))*a(n-j), j=1..n)/n)
%p A225198     end:
%p A225198 seq(a(n), n=0..45);  # _Alois P. Heinz_, Mar 15 2014
%t A225198 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_ *)
%t A225198 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 *)
%o A225198 (PARI) x='x+O('x^66); r=8; Vec( prod(k=1,r-1, (1-x^k)^(r-k)) / eta(x)^r )
%o A225198 (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
%o A225198 (Sage)
%o A225198 m=50; r=8
%o A225198 R = PowerSeriesRing(ZZ, 'x')
%o A225198 x = R.gen().O(m)
%o A225198 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 A225198 s.coefficients() # _G. C. Greubel_, Dec 10 2018
%Y A225198 A row of the array in A242641.
%Y A225198 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 A225198 nonn
%O A225198 0,3
%A A225198 _Joerg Arndt_, May 01 2013