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%I M2554 N1011 #59 Oct 27 2023 18:03:15
%S 1,1,3,6,12,21,40,67,117,193,319,510,818,1274,1983,3032,4610,6915,
%T 10324,15235,22371,32554,47119,67689,96763,137404,194211,272939,
%U 381872,531576,736923,1016904,1397853,1913561,2610023,3546507,4802694,6481101,8718309,11689929,15627591,20828892
%N Number of 3-line partitions of n.
%C Planar partitions into at most three rows. - _Joerg Arndt_, May 01 2013
%C Number of partitions of n where there is one sort of part 1, two sorts of part 2, and three sorts of every other part. - _Joerg Arndt_, Mar 15 2014
%D L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.8).
%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="/A000991/b000991.txt">Table of n, a(n) for n = 0..6000</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 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-x)^2 * (1-x^2) / Product_(k>=1, 1-x^k )^3.
%F For n>=4, a(n) = A000716(n) - 2*A000716(n-1) + 2*A000716(n-3) - A000716(n-4). - _Vaclav Kotesovec_, Oct 28 2015
%F a(n) ~ Pi^3 * exp(Pi*sqrt(2*n)) / (16*n^3). - _Vaclav Kotesovec_, Oct 28 2015
%p b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(binomial(min(i, 3)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..45); # _Alois P. Heinz_, Mar 15 2014
%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[Min[i, 3]+j-1, j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, Mar 20 2014, after _Alois P. Heinz_ *)
%t nmax = 40; CoefficientList[Series[(1-x)^2 * (1-x^2) * Product[1/(1-x^k)^3, {k,1,nmax}], {x,0,nmax}], x] (* _Vaclav Kotesovec_, Oct 28 2015 *)
%o (PARI) x='x+O('x^66); Vec((1-x)^2*(1-x^2)/eta(x)^3) \\ _Joerg Arndt_, May 01 2013
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^2*(1-x^2)/(&*[1-x^j: j in [1..2*m]])^3 )); // _G. C. Greubel_, Dec 06 2018
%o (Sage)
%o R = PowerSeriesRing(ZZ, 'x')
%o x = R.gen().O(50)
%o s = (1-x)^2 * (1-x^2) / prod(1-x^j for j in (1..60))^3
%o s.coefficients()
%o # _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 G.f. corrected by _Sean A. Irvine_, Oct 19 2011
%E G.f. corrected by _Joerg Arndt_, May 01 2013
%E Prepended a(0)=1, added more terms, _Joerg Arndt_, May 01 2013
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