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

0,3

Links

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

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).

Example

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.

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 *)

Crossrefs

Cf. A103628, A276422, A276423, A276424.

Sequence in context: A000673 A355640 A370586 * A129383 A052957 A275441

Adjacent sequences: A276422 A276423 A276424 * A276426 A276427 A276428

Keyword

nonn

Author

Emeric Deutsch, Sep 14 2016

Status

approved