A265256 -id:A265256 - cp's OEIS frontend

A276422 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its odd singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once.

1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 4, 0, 0, 0, 1, 2, 2, 0, 2, 0, 1, 8, 0, 0, 1, 1, 0, 1, 4, 4, 0, 4, 0, 2, 0, 1, 14, 0, 0, 2, 2, 1, 1, 0, 2, 9, 6, 0, 7, 0, 4, 0, 2, 0, 2, 24, 1, 0, 4, 3, 2, 2, 1, 3, 0, 2, 16, 10, 0, 12, 0, 8, 0, 4, 1, 3, 0, 2, 41, 1, 0, 7, 5, 4, 4, 2, 6, 1, 3, 0, 3, 28, 16, 0, 20, 0, 14, 0, 8, 2, 6, 1, 3, 0, 3

Offset: 0

Emeric Deutsch, Sep 14 2016

T(n,0) = A265256(n).
T(n,n) = A000700(n).
Sum(k*T(n,k), k>=0) = A276423(n).
Sum(T(n,k), k>=0) = A000041(n).

			Row 4 is 4, 0, 0, 0, 1 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.
Row 5 is 2, 2, 0, 2, 0, 1 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.
Triangle starts:
1;
0,1;
2,0,0;
1,1,0,1;
4,0,0,0,1;
2,2,0,2,0,1.

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000041, A000700, A265256, A276423, A276424, A276425.

g := Product(((1-x^(2*j-1))*(1+t^(2*j-1)*x^(2*j-1))+x^(4*j-2))/(1-x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 23)): for n from 0 to 20 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 20 do seq(coeff(P[n], t, i), i = 0 .. 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^i, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Sep 14 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^i, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Oct 04 2016, after Alois P. Heinz *)

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

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

A276422 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its odd singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once.

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