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

%I A026805 #33 Nov 02 2019 11:29:35
%S A026805 0,1,0,2,1,3,2,6,5,9,9,16,17,26,28,42,48,66,77,105,122,160,189,245,
%T A026805 290,368,436,547,650,804,954,1174,1390,1693,2004,2425,2865,3445,4060,
%U A026805 4858,5716,6802,7986,9468,11087,13088,15298,17995,20987,24604,28631,33464
%N A026805 Number of partitions of n in which the least part is even.
%C A026805 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
%H A026805 Vaclav Kotesovec, <a href="/A026805/b026805.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%F A026805 From _Vladeta Jovovic_, Aug 26 2003: (Start)
%F A026805 G.f.: Sum_{k>=2} ((-1)^k*(-1+1/Product_{i>=k} (1-x^i))).
%F A026805 a(n) = Sum_{k=2..n} (-1)^k*A026807(n, k) = A000041(n)-A026804(n). (End)
%F A026805 From _Emeric Deutsch_, Apr 04 2006: (Start)
%F A026805 G.f.: Sum_{k>=1}(x^(2k)/Product_{j>=2k}(1-x^j)).
%F A026805 G.f.: Sum_{k>=1}(x^(2k)/((1-x^(2k))*Product_{j=1..k-1}(1-x^j))). (End)
%F A026805 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
%e A026805 a(6)=3 because we have [6],[4,2] and [2,2,2].
%p A026805 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
%p A026805 # second Maple program:
%p A026805 b:= proc(n, i) option remember; `if`(n<1 or i<1, 0, b(n, i-1)+
%p A026805       `if`(n=i, 1-irem(n, 2), 0)+`if`(i>n, 0, b(n-i, i)))
%p A026805     end:
%p A026805 a:= n-> b(n$2):
%p A026805 seq(a(n), n=1..60);  # _Alois P. Heinz_, Jul 26 2015
%t A026805 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_ *)
%K A026805 nonn
%O A026805 1,4
%A A026805 _Clark Kimberling_