%I M2563 N1014 #62 Sep 08 2022 08:44:31

%S 1,1,3,6,13,23,45,78,141,239,409,674,1116,1794,2882,4544,7131,11031,

%T 16983,25844,39124,58680,87538,129578,190830,279140,406334,588026,

%U 847034,1213764,1731780,2459244,3478185,4898285,6872041,9603356,13372607,18553871,25656865

%N Number of 4-line partitions of n (i.e., planar partitions of n with at most 4 lines).

%C Number of partitions of n where there is one sort of part 1, two sorts of part 2, three sorts of part 3, and four sorts of every other part. - _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 Alois P. Heinz and Vaclav Kotesovec, <a href="/A002799/b002799.txt">Table of n, a(n) for n = 0..10000</a> (terms 1..1000 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="/A002799/a002799.jpg">Graph - The asymptotic ratio (35000 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.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F Euler transform of 1, 2, 3, 4, 4, 4, ...

%F G.f.: (1-x)^3 * (1-x^2)^2 * (1-x^3) / Product_{k>=1} (1-x^k)^4. - _Joerg Arndt_, May 01 2013

%F a(n) ~ 2^(13/4) * Pi^6 * exp(2*Pi*sqrt(2*n/3)) / (3^(13/4) * n^(19/4)). - _Vaclav Kotesovec_, Oct 28 2015

%p with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> `if`(n<5,n,4)): seq(a(n), n=0..40); # _Alois P. Heinz_, Sep 08 2008

%t etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a = etr[Min[#, 4]&]; Join[{1}, Table[a[n], {n, 1, 38}]] (* _Jean-François Alcover_, Mar 10 2014, after _Alois P. Heinz_ *)

%t nmax = 40; CoefficientList[Series[(1-x)^3 * (1-x^2)^2 * (1-x^3) * Product[1/(1-x^k)^4, {k,1,nmax}], {x,0,nmax}], x] (* _Vaclav Kotesovec_, Oct 28 2015 *)

%o (PARI) x='x+O('x^66); r=4; 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)^3*(1-x^2)^2*(1-x^3)/(&*[1-x^j: j in [1..2*m]] )^4 )); // _G. C. Greubel_, Dec 06 2018

%o (Sage)

%o R = PowerSeriesRing(ZZ, 'x')

%o x = R.gen().O(50)

%o s = (1-x)^3*(1-x^2)^2*(1-x^3)/prod(1-x^j for j in (1..60))^4

%o s.coefficients() # _G. C. Greubel_, Dec 06 2018

%Y A row of the array in A242641.

%Y Cf. A000219, A001452.

%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 Edited and extended with formula by _Christian G. Bower_, Jan 01 2004

%E a(0)=1 prepended by _Joerg Arndt_, May 01 2013

%E Offset corrected by _Vaclav Kotesovec_, Oct 28 2015