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A026837 Number of partitions of n into distinct parts, the greatest being odd. 7
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 (list; graph; refs; listen; history; text; internal format)
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 * A005855 A096748 A263659

Adjacent sequences: A026834 A026835 A026836 * A026838 A026839 A026840

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified December 3 09:03 EST 2022. Contains 358515 sequences. (Running on oeis4.)