|
%I M2564 N1015 #53 Oct 27 2023 19:13:34
%S 1,1,3,6,13,24,47,83,152,263,457,768,1292,2118,3462,5564,8888,14016,
%T 21973,34081,52552,80331,122078,184161,276303,411870,610818,900721,
%U 1321848,1929981,2805338,4058812,5847966,8390097,11990531,17069145,24210571,34215537,48190451,67644522
%N Number of 5-line partitions of n.
%C Planar partitions into at most five rows. - _Joerg Arndt_, May 01 2013
%C Number of partitions of n where there are k sorts of parts k for k<=4 and 5 sorts all other parts. - _Joerg Arndt_, Mar 15 2014
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vaclav Kotesovec, <a href="/A001452/b001452.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from Alois P. Heinz)
%H M. S. Cheema and B. Gordon, <a href="http://dx.doi.org/10.1215/S0012-7094-64-03125-4">Some remarks on two- and three-line partitions</a>, Duke Math. J., 31 (1964), 267-273.
%H Vaclav Kotesovec, <a href="/A001452/a001452.jpg">Graph - The asymptotic ratio (50000 terms)</a>
%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.
%F G.f.: 1 / Product_{k>=1} (1-x^k)^min(k,5). - _Sean A. Irvine_, Jul 24 2012
%F a(n) ~ 15625 * Pi^10 * sqrt(5) * exp(Pi*sqrt(10*n/3)) / (2592 * sqrt(3) * n^7). - _Vaclav Kotesovec_, Oct 28 2015
%p with(numtheory):
%p a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p min(d, 5)*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, 5]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, Mar 17 2014, after _Alois P. Heinz_ *)
%t nmax = 40; CoefficientList[Series[(1-x)^4 * (1-x^2)^3 * (1-x^3)^2 * (1-x^4) * Product[1/(1-x^k)^5, {k,1,nmax}], {x,0,nmax}], x] (* _Vaclav Kotesovec_, Oct 28 2015 *)
%o (PARI) x='x+O('x^66); r=5; Vec( prod(k=1,r-1, (1-x^k)^(r-k)) / eta(x)^r ) \\ _Joerg Arndt_, May 01 2013
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)/(&*[1-x^j: j in [1..2*m]])^5 )); // _G. C. Greubel_, Dec 06 2018
%o (Sage)
%o R = PowerSeriesRing(ZZ, 'x')
%o x = R.gen().O(50)
%o s = (1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)/prod(1-x^j for j in (1..60))^5
%o list(s) # _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 _N. J. A. Sloane_
%E More terms from _Sean A. Irvine_, Jul 24 2012
%E a(0)=1 prepended by _Joerg Arndt_, May 01 2013
|
