A026805 Number of partitions of n in which the least part is even.

0, 1, 0, 2, 1, 3, 2, 6, 5, 9, 9, 16, 17, 26, 28, 42, 48, 66, 77, 105, 122, 160, 189, 245, 290, 368, 436, 547, 650, 804, 954, 1174, 1390, 1693, 2004, 2425, 2865, 3445, 4060, 4858, 5716, 6802, 7986, 9468, 11087, 13088, 15298, 17995, 20987, 24604, 28631, 33464

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Also number of partitions of n in which the largest part occurs an even number of times. Example: a(6)=3 because we have [3,3],[2,2,1,1] and [1,1,1,1,1,1]. - Emeric Deutsch, Apr 04 2006

Examples

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

Links

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

Programs

Maple

g:=sum(x^(2*k)/(1-x^(2*k))/product(1-x^j,j=1..k-1),k=1..40): gser:=series(g,x=0,52): seq(coeff(gser,x,n),n=1..49); # 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, 1-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, 1-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 28 2015, after Alois P. Heinz *)

Formula

From Vladeta Jovovic, Aug 26 2003: (Start)

G.f.: Sum_{k>=2} ((-1)^k*(-1+1/Product_{i>=k} (1-x^i))).

a(n) = Sum_{k=2..n} (-1)^k*A026807(n, k) = A000041(n)-A026804(n). (End)

From Emeric Deutsch, Apr 04 2006: (Start)

G.f.: Sum_{k>=1}(x^(2k)/Product_{j>=2k}(1-x^j)).

G.f.: Sum_{k>=1}(x^(2k)/((1-x^(2k))*Product_{j=1..k-1}(1-x^j))). (End)

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 61*Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Jul 06 2019, extended Nov 02 2019