OFFSET
1,8
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (first 70 terms from Vincenzo Librandi)
FORMULA
G.f.: (x^3/(1 + x^3)) * Product_{j >= 1} (1 + x^j). - Emeric Deutsch, Apr 17 2006
Corresponding g.f. for "number of k's" is (x^k/(1 + x^k)) * Product_{j >= 1} (1 + x^j). - Joerg Arndt, Feb 20 2014
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015
EXAMPLE
a(9) = 3 because in the eight partitions of 9 into distinct parts, namely [9], [8, 1], [7, 2], [6, 3], [6, 2, 1], [5, 4], [5, 3, 1] and [4, 3, 2], only three contain 3.
MAPLE
g:=x^3*product(1+x^j, j=1..60)/(1+x^3): gser:=series(g, x=0, 57): seq(coeff(gser, x, n), n=1..54); # Emeric Deutsch, Apr 17 2006
MATHEMATICA
Table[Count[Select[IntegerPartitions[n], Length[Union[#]] == Length[#] &], _?(MemberQ[#, 3] &)], {n, 60}] (* Harvey P. Dale, Aug 19 2011 *)
nmax = 100; Rest[CoefficientList[Series[x^3/(1 + x^3) * Product[1 + x^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 30 2015 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Example clarified by Harvey P. Dale, Aug 19 2011
STATUS
approved