A265257 -id:A265257 - cp's OEIS frontend

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

Emeric Deutsch, Sep 14 2016

nonn

			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.

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

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

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

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 *)

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

A265255 Triangle read by rows: T(n,k) is the number of partitions of n having k odd singletons (n, k >=0).

Original entry on oeis.org

1, 0, 1, 2, 1, 2, 4, 0, 1, 2, 5, 8, 1, 2, 4, 11, 14, 3, 5, 9, 20, 0, 1, 24, 8, 10, 16, 37, 1, 2, 41, 15, 21, 28, 65, 3, 5, 66, 30, 39, 49, 108, 9, 10, 104, 57, 69, 0, 1, 80, 178, 19, 20, 163, 99, 120, 1, 2, 128, 286, 39, 37, 248, 170, 201, 3, 5, 203, 448, 73, 68, 372, 284, 327

Offset: 0

Emeric Deutsch, Jan 01 2016

Sum of entries in row n is A000041(n).
T(n,0) = A265256(n).
Sum_{k>=0} k*T(n,k) = A265257(n).

			T(6,2) = 2 because each of the partitions [1,2,3], [1,5] of n = 6 has 2 odd singleton, while the other partitions, namely [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2], [2,2,2], [1,1,1,3], [3,3], [1,1,4], [2,4], [6], have 0, 0, 0, 0, 1, 0, 0, 0, 0  odd singletons.
Triangle starts:
1;
0, 1;
2;
1, 2;
4, 0, 1;
2, 5;
8, 1, 2.

Alois P. Heinz, Rows n = 0..1000, flattened

Cf. A000041, A265253, A265256, A265257.

g := mul(((1-x^(2*j-1))*(1+t*x^(2*j-1))+x^(4*j-2))/(1-x^j), j = 1 .. 80): gser := simplify(series(g, x = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, q), q = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
      `if`(j=1 and i::odd, x, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..30);  # Alois P. Heinz, Jan 01 2016

b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*If[j == 1 && OddQ[i], x, 1], {j, 0, n/i}]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 10 2016, after Alois P. Heinz *)

G.f.: G(t,x) = Product_{j>=1} ((1 -x^(2j-1))(1+tx^{2j-1}) + x^(4j-2))/ (1-x^j).

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula