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%I #26 May 25 2018 13:10:58
%S 0,1,0,4,4,13,13,33,41,79,98,171,223,354,458,692,905,1306,1694,2375,
%T 3077,4202,5401,7238,9260,12200,15495,20145,25446,32686,41020,52170,
%U 65117,82071,101852,127374,157277,195289,239915,296023,362000,444063,540595,659662
%N Sum of the odd singletons in all partitions of n (n>=0). A singleton in a partition is a part that occurs exactly once.
%H Alois P. Heinz, <a href="/A276423/b276423.txt">Table of n, a(n) for n = 0..2000</a>
%F G.f.: g(x) = x*(1-x+3*x^2+3*x^4-x^5+x^6)/((1-x^4)^2*Product_{j>=1} 1-x^j).
%F a(n) = Sum_{k>=0} k*A276422(n,k).
%e a(4) = 4 because in the partitions [1,1,1,1], [1,1,2], [2,2], [1,3], [4] the sums of the odd singletons are 0,0,0,4,0, respectively; their sum is 4.
%e a(5) = 13 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5] the sums of the odd singletons are 0,0,1,3,3,1,5, respectively; their sum is 13.
%p g := x*(1-x+3*x^2+3*x^4-x^5+x^6)/((1-x^4)^2*(product(1-x^i, i = 1..120))): gser := series(g, x = 0, 60); seq(coeff(gser, x, n), n = 0..50);
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p `if`(i<1, 0, add((p-> p+`if`(i::odd and j=1,
%p [0, i*p[1]], 0))(b(n-i*j, i-1)), j=0..n/i)))
%p end:
%p a:= n-> b(n$2)[2]:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Sep 14 2016
%t b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, 0, Sum[Function[p, p + If[OddQ[i] && j == 1, {0, If[p === 0, 0, i*p[[1]]]}, 0]][b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Dec 04 2016 after _Alois P. Heinz_ *)
%t Table[Total[Select[Flatten[Tally/@IntegerPartitions[n],1],#[[2]]==1 && OddQ[ #[[1]]]&][[All,1]]],{n,0,50}] (* _Harvey P. Dale_, May 25 2018 *)
%Y Cf. A103628, A265257, A276422, A276424, A276425.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Sep 14 2016
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