OFFSET
1,5
COMMENTS
Fine's theorem: A026838(n) - a(n) = 1 if n = k(3k+1)/2, = -1 if n = k(3k-1)/2, = 0 otherwise (see A143062).
Also number of partitions of n into an odd number of parts and such that parts of every size from 1 to the largest occur. Example: a(9)=4 because we have [3,2,2,1,1],[2,2,2,2,1],[2,2,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 04 2006
LINKS
I. Pak, On Fine's partition theorems, Dyson, Andrews and missed opportunities, Math. Intelligencer, 25 (No. 1, 2003), 10-16.
FORMULA
G.f.: sum(k>=1, x^(2k-1) * prod(j=1..2k-2, 1+x^j ) ). - Emeric Deutsch, Apr 04 2006
a(2*n) = A118302(2*n), a(2*n-1) = A118301(2*n-1); a(n) = A000009(n) - A026838(n). - Reinhard Zumkeller, Apr 22 2006
EXAMPLE
a(9)=4 because we have [9],[7,2],[5,4] and [5,3,1].
MAPLE
g:=sum(x^(2*k-1)*product(1+x^j, j=1..2*k-2), k=1..40): gser:=series(g, x=0, 60): seq(coeff(g, x, n), n=1..54); # Emeric Deutsch, Apr 04 2006
MATHEMATICA
Table[Count[IntegerPartitions[n], _?(Length[#]==Length[Union[#]] && OddQ[ First[#]]&)], {n, 60}] (* Harvey P. Dale, Jun 28 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved