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A116676 Number of odd parts in all partitions of n into distinct parts.

%I A116676 #18 May 24 2022 18:25:37
%S A116676 0,1,0,2,2,3,4,5,8,10,14,16,22,26,34,43,54,64,80,96,116,142,170,202,
%T A116676 242,288,340,404,474,556,652,762,886,1034,1198,1389,1606,1852,2132,
%U A116676 2454,2814,3224,3690,4214,4804,5478,6228,7072,8028,9094,10290,11635,13134
%N A116676 Number of odd parts in all partitions of n into distinct parts.
%C A116676 a(n) = Sum(k*A116675(n,k), k>=0).
%H A116676 Vaclav Kotesovec, <a href="/A116676/b116676.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A116676 G.f.: product(1+x^j, j=1..infinity)*sum(x^(2j-1)/(1+x^(2j-1)), j=1..infinity).
%F A116676 For n > 0, a(n) = A015723(n) - A116680(n). - _Vaclav Kotesovec_, May 26 2018
%F A116676 a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - _Vaclav Kotesovec_, May 26 2018
%e A116676 a(9) = 10 because in the partitions of 9 into distinct parts, namely, [9], [81], [72], [6,3], [6,2,1], [5,4], [5,3,1] and [4,3,2], we have a total of 10 odd parts.
%p A116676 f:=product(1+x^j,j=1..64)*sum(x^(2*j-1)/(1+x^(2*j-1)),j=1..35): fser:=series(f,x=0,60): seq(coeff(fser,x,n),n=0..56);
%p A116676 # second Maple program:
%p A116676 b:= proc(n, i) option remember; local f, g;
%p A116676       if n=0 then [1, 0] elif i<1 then [0, 0]
%p A116676     else f:=b(n, i-1); g:=`if`(i>n, [0, 0], b(n-i, min(n-i, i-1)));
%p A116676          [f[1]+g[1], f[2]+g[2] +irem(i, 2)*g[1]]
%p A116676       fi
%p A116676     end:
%p A116676 a:= n-> b(n, n)[2]:
%p A116676 seq(a(n), n=0..60);  # _Alois P. Heinz_, Nov 21 2012
%t A116676 b[n_, i_] := b[n, i] = Module[{f, g}, Which [n == 0, {1, 0}, i<1 , {0, 0}, True, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, Min[n-i, i-1]]]; {f[[1]] + g[[1]],       f[[2]] + g[[2]] + Mod[i, 2]*g[[1]]}]]; a[n_] := b[n, n][[2]]; Table [a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 22 2015, after _Alois P. Heinz_ *)
%Y A116676 Cf. A116675, A305123, A305124.
%Y A116676 Cf. A305082, A015723, A090867, A116680, A067588.
%K A116676 nonn
%O A116676 0,4
%A A116676 _Emeric Deutsch_, Feb 22 2006