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A026838 Number of partitions of n into distinct parts, the greatest being even. 8
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 10 21:08 EDT 2021. Contains 343780 sequences. (Running on oeis4.)