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

1,4

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

Links

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

Example

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

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 *)

Crossrefs

Sequence in context: A337452 A164768 A006208 * A335688 A206494 A022477

Adjacent sequences: A026802 A026803 A026804 * A026806 A026807 A026808

Keyword

nonn

Author

Clark Kimberling

Status

approved