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A001452
Number of 5-line partitions of n.
(Formerly M2564 N1015)
11
1, 1, 3, 6, 13, 24, 47, 83, 152, 263, 457, 768, 1292, 2118, 3462, 5564, 8888, 14016, 21973, 34081, 52552, 80331, 122078, 184161, 276303, 411870, 610818, 900721, 1321848, 1929981, 2805338, 4058812, 5847966, 8390097, 11990531, 17069145, 24210571, 34215537, 48190451, 67644522
OFFSET
0,3
COMMENTS
Planar partitions into at most five rows. - Joerg Arndt, May 01 2013
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
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
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (first 1000 terms 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.
FORMULA
G.f.: 1 / Product_{k>=1} (1-x^k)^min(k,5). - Sean A. Irvine, Jul 24 2012
a(n) ~ 15625 * Pi^10 * sqrt(5) * exp(Pi*sqrt(10*n/3)) / (2592 * sqrt(3) * n^7). - Vaclav Kotesovec, Oct 28 2015
MAPLE
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
min(d, 5)*d, d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..45); # Alois P. Heinz, Mar 15 2014
MATHEMATICA
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 *)
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 *)
PROG
(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
(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
(Sage)
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
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
list(s) # G. C. Greubel, Dec 06 2018
CROSSREFS
A row of the array in A242641.
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: A293076 A293421 A018081 * A005405 A225196 A301597
KEYWORD
nonn
EXTENSIONS
More terms from Sean A. Irvine, Jul 24 2012
a(0)=1 prepended by Joerg Arndt, May 01 2013
STATUS
approved