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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

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

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).

EXAMPLE

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.

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-Fran├žois 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 *)

CROSSREFS

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

Sequence in context: A183362 A088838 A127403 * A052993 A214779 A323920

Adjacent sequences:  A276420 A276421 A276422 * A276424 A276425 A276426

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 14 2016

STATUS

approved

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Last modified October 14 18:28 EDT 2019. Contains 328022 sequences. (Running on oeis4.)