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
KEYWORD
nonn
AUTHOR
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