A002799 Number of 4-line partitions of n (i.e., planar partitions of n with at most 4 lines).
1, 1, 3, 6, 13, 23, 45, 78, 141, 239, 409, 674, 1116, 1794, 2882, 4544, 7131, 11031, 16983, 25844, 39124, 58680, 87538, 129578, 190830, 279140, 406334, 588026, 847034, 1213764, 1731780, 2459244, 3478185, 4898285, 6872041, 9603356, 13372607, 18553871, 25656865
Offset: 0
Keywords
References
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Alois P. Heinz)
- M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273.
- Vaclav Kotesovec, Graph - The asymptotic ratio (35000 terms)
- P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116.
- N. J. A. Sloane, Transforms
Crossrefs
Programs
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Magma
m:=50; R
:=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 -
Maple
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
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Mathematica
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 *) 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 *) -
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 -
Sage
R = PowerSeriesRing(ZZ, 'x') x = R.gen().O(50) s = (1-x)^3*(1-x^2)^2*(1-x^3)/prod(1-x^j for j in (1..60))^4 s.coefficients() # G. C. Greubel, Dec 06 2018
Formula
Euler transform of 1, 2, 3, 4, 4, 4, ...
G.f.: (1-x)^3 * (1-x^2)^2 * (1-x^3) / Product_{k>=1} (1-x^k)^4. - Joerg Arndt, May 01 2013
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
Extensions
Edited and extended with formula by Christian G. Bower, Jan 01 2004
a(0)=1 prepended by Joerg Arndt, May 01 2013
Offset corrected by Vaclav Kotesovec, Oct 28 2015
Comments