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

Author

Clark Kimberling

Keywords

• nonn

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

Examples

			a(9)=4 because we have [9],[7,2],[5,4] and [5,3,1].
		

Links

• I. Pak, On Fine's partition theorems, Dyson, Andrews and missed opportunities, Math. Intelligencer, 25 (No. 1, 2003), 10-16.

Crossrefs

Cf. A026838.

Cf. A027193.

Programs

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 *)

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