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%I #32 Nov 02 2019 11:31:43
%S 1,1,3,3,6,8,13,16,25,33,47,61,84,109,148,189,249,319,413,522,670,842,
%T 1066,1330,1668,2068,2574,3171,3915,4800,5888,7175,8753,10617,12879,
%U 15552,18772,22570,27125,32480,38867,46372,55275,65707,78047,92470,109456
%N Number of partitions of n in which the least part is odd.
%C 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
%H Vaclav Kotesovec, <a href="/A026804/b026804.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%F 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
%F G.f.: Sum_{j>=1}(x^j/(1+x^j)/Product_{i=1..j}(1-x^i)). - _Vladeta Jovovic_, Aug 11 2004
%F G.f.: Sum_{k>=1}(x^(2k-1)/Product_{j>=2k-1}(1-x^j)). - _Emeric Deutsch_, Apr 04 2006
%F 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
%e 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).
%p 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
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(n<1 or i<1, 0, b(n, i-1)+
%p `if`(n=i, irem(n, 2), 0)+`if`(i>n, 0, b(n-i, i)))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=1..60); # _Alois P. Heinz_, Jul 26 2015
%t 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_ *)
%o (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)));
%o a(n) = b(n, n); \\ _Indranil Ghosh_, Jun 22 2017, after Maple code by _Alois P. Heinz_
%Y Cf. A046746.
%K nonn
%O 1,3
%A _Clark Kimberling_
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