A026838 Number of partitions of n into distinct parts, the greatest being even.

0, 1, 1, 1, 1, 2, 3, 3, 4, 5, 6, 7, 9, 11, 14, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 83, 96, 111, 128, 148, 170, 195, 224, 256, 292, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2048, 2291, 2560, 2859

Offset

1,6

Comments

Fine's theorem: a(n) - A026837(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 even number of parts and such that parts of every size from 1 to the largest occur. Example: a(8)=3 because we have [3,2,2,1], [2,2,1,1,1,1] and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 04 2006

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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) * prod(j=1..2k-1, 1+x^j ) ). - Emeric Deutsch, Apr 04 2006

a(2*n) = A118301(2*n), a(2*n-1) = A118302(2*n-1); a(n) = A000009(n) - A026837(n). - Reinhard Zumkeller, Apr 22 2006

Example

a(8)=3 because we have [8],[6,2] and [4,3,1].

Maple

g:=sum(x^(2*k)*product(1+x^j, j=1..2*k-1), k=1..50): gser:=series(g, x=0, 75): seq(coeff(gser, x, n), n=1..54); # Emeric Deutsch, Apr 04 2006

Mathematica

nn=54; CoefficientList[Series[Sum[x^(2j)Product[1+ x^i, {i, 1, 2j-1}], {j, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Jun 20 2014 *)

Crossrefs

Cf. A026837, A027187.

Sequence in context: A029036 A192530 A281744 * A182229 A017864 A188937

Adjacent sequences: A026835 A026836 A026837 * A026839 A026840 A026841

Keyword

nonn

Author

Clark Kimberling

Status

approved