A276424 Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its even singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once.

1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 0, 1, 4, 0, 2, 0, 1, 0, 6, 0, 2, 0, 1, 0, 2, 8, 0, 3, 0, 2, 0, 2, 0, 11, 0, 4, 0, 3, 0, 2, 0, 2, 15, 0, 5, 0, 4, 0, 4, 0, 2, 0, 19, 0, 7, 0, 6, 0, 5, 0, 2, 0, 3, 25, 0, 9, 0, 8, 0, 7, 0, 4, 0, 3, 0, 34, 0, 11, 0, 10, 0, 10, 0, 5, 0, 3, 0, 4

Offset

0,7

Comments

T(n,0) = A265254(n).

T(n,n) = A035457(n).

Sum_{k>=0} k*T(n,k) = A276425(n).

Sum_{k>=0} T(n,k) = A000041(n).

Links

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

Formula

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

Example

Row 4 is 3, 0, 1, 0, 1 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.
Row 5 is 4, 0, 2, 0, 1, 0 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.
Triangle starts:
1;
1,0;
1,0,1;
2,0,1,0;
3,0,1,0,1;
4,0,2,0,1,0;
6,0,2,0,1,0,2.

Maple

g := Product(((1-x^(2*j))*(1+t^(2*j)*x^(2*j))+x^(4*j))/(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::even, 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

Mathematica

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 && EvenQ[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, Dec 11 2016, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A035457, A265254, A276422, A276423, A276425.

Sequence in context: A065334 A356945 A162590 * A346070 A191258 A254990

Adjacent sequences: A276421 A276422 A276423 * A276425 A276426 A276427

Keyword

nonn,tabl

Author

Emeric Deutsch, Sep 14 2016

Status

approved