A026837 Number of partitions of n into distinct parts, the greatest being odd.

1, 0, 1, 1, 2, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 16, 19, 23, 27, 32, 38, 45, 52, 61, 71, 82, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 556, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2049, 2291, 2560, 2859

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

Table of n, a(n) for n=1..54.

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

Cf. A026838.

Cf. A027193.

Sequence in context: A256636 A258327 A102240 * A366916 A005855 A096748

Adjacent sequences: A026834 A026835 A026836 * A026838 A026839 A026840

Keyword

nonn

Author

Clark Kimberling

Status

approved