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A026804 Number of partitions of n in which the least part is odd. 7
1, 1, 3, 3, 6, 8, 13, 16, 25, 33, 47, 61, 84, 109, 148, 189, 249, 319, 413, 522, 670, 842, 1066, 1330, 1668, 2068, 2574, 3171, 3915, 4800, 5888, 7175, 8753, 10617, 12879, 15552, 18772, 22570, 27125, 32480, 38867, 46372, 55275, 65707, 78047, 92470, 109456 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Also number of partitions of n in which the largest part occurs an odd number of times. Example: a(5)=6 because we have [5],[4,1],[3,2],[3,1,1],[2,1,1,1] and [1,1,1,1,1] ([2,2,1] does not qualify). - Emeric Deutsch, Apr 04 2006

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

FORMULA

G.f.: Sum_{k>=1}((-1)^(k+1)*(-1+1/Product_{i>=k} (1-x^i))). a(n) = Sum_{k=1..n}(-1)^(k+1)*A026807(n, k). - Vladeta Jovovic, Aug 26 2003

G.f.: Sum_{j>=1}(x^j/(1+x^j)/Product_{i=1..j}(1-x^i)). - Vladeta Jovovic, Aug 11 2004

G.f.: Sum_{k>=1}(x^(2k-1)/Product_{j>=2k-1}(1-x^j)). - Emeric Deutsch, Apr 04 2006

EXAMPLE

a(5)=6 because we have [5],[4,1],[3,1,1],[2,2,1],[2,1,1,1] and [1,1,1,1,1] ([3,2] does not qualify).

MAPLE

g:=sum(x^(2*k-1)/product(1-x^j, j=2*k-1..50), k=1..50): gser:=series(g, x=0, 45): seq(coeff(gser, x, n), n=1..43); # Emeric Deutsch, Apr 04 2006

# second Maple program:

b:= proc(n, i) option remember; `if`(n<1 or i<1, 0, b(n, i-1)+

      `if`(n=i, irem(n, 2), 0)+`if`(i>n, 0, b(n-i, i)))

    end:

a:= n-> b(n$2):

seq(a(n), n=1..60);  # Alois P. Heinz, Jul 26 2015

MATHEMATICA

b[n_, i_] := b[n, i] = If[n < 1 || i < 1, 0, b[n, i - 1] + If[n == i, Mod[n, 2], 0] + If[i > n, 0, b[n - i, i]]]; a[n_] :=  b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-Fran├žois Alcover, Oct 09 2015, after Alois P. Heinz *)

PROG

(PARI) b(n, i) = if(n<1 || i<1, 0, b(n, i - 1) + if(n==i, n%2 , 0) + if(i>n, 0, b(n - i, i)));

a(n) = b(n, n); \\ Indranil Ghosh, Jun 22 2017, after Maple code by Alois P. Heinz

CROSSREFS

Cf. A046746.

Sequence in context: A143592 A280197 A097307 * A240213 A205970 A104715

Adjacent sequences:  A026801 A026802 A026803 * A026805 A026806 A026807

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified November 14 09:49 EST 2018. Contains 317182 sequences. (Running on oeis4.)