A002799 Number of 4-line partitions of n (i.e., planar partitions of n with at most 4 lines). (Formerly M2563 N1014)

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

0,3

Comments

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

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

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

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

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 *)

Prog

(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
(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
(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

Crossrefs

A row of the array in A242641.

Cf. A000219, A001452.

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).

Sequence in context: A022811 A295730 A323580 * A285263 A162426 A058554

Adjacent sequences: A002796 A002797 A002798 * A002800 A002801 A002802

Keyword

nonn

Author

N. J. A. Sloane

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

Status

approved