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A276423 Sum of the odd singletons in all partitions of n (n>=0). A singleton in a partition is a part that occurs exactly once.

%I A276423 #28 Jun 12 2025 02:18:39
%S A276423 0,1,0,4,4,13,13,33,41,79,98,171,223,354,458,692,905,1306,1694,2375,
%T A276423 3077,4202,5401,7238,9260,12200,15495,20145,25446,32686,41020,52170,
%U A276423 65117,82071,101852,127374,157277,195289,239915,296023,362000,444063,540595,659662
%N A276423 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 A276423 Alois P. Heinz, <a href="/A276423/b276423.txt">Table of n, a(n) for n = 0..2000</a>
%F A276423 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 A276423 a(n) = Sum_{k>=0} k*A276422(n,k).
%F A276423 a(n) ~ 3^(3/2) * exp(Pi*sqrt(2*n/3)) / (16*Pi^2). - _Vaclav Kotesovec_, Jun 12 2025
%e A276423 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 A276423 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 A276423 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 A276423 # second Maple program:
%p A276423 b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p A276423       `if`(i<1, 0, add((p-> p+`if`(i::odd and j=1,
%p A276423       [0, i*p[1]], 0))(b(n-i*j, i-1)), j=0..n/i)))
%p A276423     end:
%p A276423 a:= n-> b(n$2)[2]:
%p A276423 seq(a(n), n=0..50); # _Alois P. Heinz_, Sep 14 2016
%t A276423 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 A276423 Table[Total[Select[Flatten[Tally/@IntegerPartitions[n],1],#[[2]]==1 && OddQ[ #[[1]]]&][[All,1]]],{n,0,50}] (* _Harvey P. Dale_, May 25 2018 *)
%Y A276423 Cf. A103628, A265257, A276422, A276424, A276425.
%K A276423 nonn
%O A276423 0,4
%A A276423 _Emeric Deutsch_, Sep 14 2016