A276425 Sum of the even singletons in all partitions of n (n>=0). A singleton in a partition is a part that occurs exactly once.

0, 0, 2, 2, 6, 8, 20, 26, 48, 66, 114, 154, 240, 326, 490, 656, 940, 1252, 1752, 2306, 3142, 4104, 5500, 7114, 9372, 12030, 15656, 19932, 25628, 32402, 41270, 51816, 65400, 81608, 102226, 126800, 157698, 194550, 240454, 295110, 362600, 442902, 541342, 658230

Offset: 0

Author

Emeric Deutsch, Sep 14 2016

Keywords

• nonn

Examples

			a(4) = 6 because in the partitions [1,1,1,1], [1,1,2], [2,2], [1,3], [4] the sums of the even singletons are 0, 2, 0, 0, 4, respectively; their sum is 6.
a(5) = 8 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 even singletons are 0, 2, 0, 0, 2, 4, 0 respectively; their sum is 8.
		

Links

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

Crossrefs

Cf. A103628, A276422, A276423, A276424.

Programs

Maple

g := 2*x^2*(1+x^2+x^4)/((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::even 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[EvenQ[i] && j == 1, {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, Oct 24 2016, after Alois P. Heinz *)

Formula

G.f.: g(x) = 2x^2*(1+x^2+x^4)/((1-x^4)^2 product(1-x^j, j>=1)).

a(n) = Sum(k*A276424(n,k), k>=0).

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