A026804 Number of partitions of n in which the least part is odd.

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

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

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

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 25*Pi/(24*sqrt(6))) / sqrt(n) + (25/16 + 2929*Pi^2/6912)/n). - Vaclav Kotesovec, Jul 06 2019, extended Nov 02 2019

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-François 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: A333526 A097307 A323435 * A240213 A205970 A104715

Adjacent sequences: A026801 A026802 A026803 * A026805 A026806 A026807

Keyword

nonn

Author

Clark Kimberling

Status

approved