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.

0, 1, 0, 4, 4, 13, 13, 33, 41, 79, 98, 171, 223, 354, 458, 692, 905, 1306, 1694, 2375, 3077, 4202, 5401, 7238, 9260, 12200, 15495, 20145, 25446, 32686, 41020, 52170, 65117, 82071, 101852, 127374, 157277, 195289, 239915, 296023, 362000, 444063, 540595, 659662

Offset: 0

Author

Emeric Deutsch, Sep 14 2016

Keywords

• nonn

Examples

			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.
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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A103628, A265257, A276422, A276424, A276425.

Programs

Maple

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);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, add((p-> p+`if`(i::odd and j=1,
      [0, i*p[1]], 0))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50); # Alois P. Heinz, Sep 14 2016

Mathematica

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 *)
Table[Total[Select[Flatten[Tally/@IntegerPartitions[n],1],#[[2]]==1 && OddQ[ #[[1]]]&][[All,1]]],{n,0,50}] (* Harvey P. Dale, May 25 2018 *)

Formula

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).

a(n) = Sum_{k>=0} k*A276422(n,k).

a(n) ~ 3^(3/2) * exp(Pi*sqrt(2*n/3)) / (16*Pi^2). - Vaclav Kotesovec, Jun 12 2025