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A225196 Number of 6-line partitions of n (i.e., planar partitions of n with at most 6 lines).

%I A225196 #43 Sep 08 2022 08:46:04
%S A225196 1,1,3,6,13,24,48,85,157,274,481,816,1388,2298,3798,6170,9968,15895,
%T A225196 25209,39550,61703,95431,146757,224036,340189,513233,770415,1149933,
%U A225196 1708277,2524846,3715285,5441762,7937671,11529512,16681995,24043245,34527521,49404590,70452001,100128249
%N A225196 Number of 6-line partitions of n (i.e., planar partitions of n with at most 6 lines).
%C A225196 Number of partitions of n where there are k sorts of parts k for k<=5 and six sorts of all other parts. - _Joerg Arndt_, Mar 15 2014
%H A225196 Vincenzo Librandi and Joerg Arndt and Alois P. Heinz, <a href="/A225196/b225196.txt">Table of n, a(n) for n = 0..1000</a>
%H A225196 Vaclav Kotesovec, <a href="/A225196/a225196.jpg">Graph - The asymptotic ratio</a> (50000 terms, convergence is slow)
%H A225196 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 A225196 G.f.: 1/Product_{n>=1} (1-x^n)^min(n,6). - _Joerg Arndt_, Mar 15 2014
%F A225196 a(n) ~ 2160 * Pi^15 * exp(2*Pi*sqrt(n)) / n^(39/4). - _Vaclav Kotesovec_, Oct 28 2015
%F A225196 G.f.: (1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/( Prod_{j>=1} (1-x^j ) )^6. - _G. C. Greubel_, Dec 06 2018
%p A225196 with(numtheory):
%p A225196 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A225196       min(d, 6)*d, d=divisors(j))*a(n-j), j=1..n)/n)
%p A225196     end:
%p A225196 seq(a(n), n=0..45);  # _Alois P. Heinz_, Mar 15 2014
%t A225196 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Min[d, 6]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, Feb 18 2015, _Alois P. Heinz_ *)
%t A225196 m:=50; CoefficientList[Series[(1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/( Product[(1-x^j), {j,1,m}])^6, {x,0,m}],x] (* _G. C. Greubel_, Dec 06 2018 *)
%o A225196 (PARI) x='x+O('x^66); r=6; Vec( prod(k=1,r-1, (1-x^k)^(r-k)) / eta(x)^r )
%o A225196 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/(&*[1-x^j: j in [1..2*m]] )^6 )); // _G. C. Greubel_, Dec 06 2018
%o A225196 (Sage)
%o A225196 R = PowerSeriesRing(ZZ,'x')
%o A225196 x = R.gen().O(50)
%o A225196 s = (1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/prod(1-x^j for j in (1..60))^6
%o A225196 s.coefficients() # _G. C. Greubel_, Dec 06 2018
%Y A225196 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).
%Y A225196 A row of the array in A242641.
%K A225196 nonn
%O A225196 0,3
%A A225196 _Joerg Arndt_, May 01 2013