A264403 Triangle read by rows: T(n,k) is the number of partitions of n in which the sum of the parts of multiplicity 1 is equal to k (0<=k<=n).

1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 2, 0, 1, 0, 2, 1, 1, 1, 1, 0, 3, 4, 0, 1, 1, 1, 0, 4, 2, 2, 1, 2, 1, 2, 0, 5, 6, 0, 2, 1, 3, 2, 2, 0, 6, 5, 2, 1, 4, 1, 4, 2, 3, 0, 8, 9, 1, 3, 2, 5, 2, 4, 3, 3, 0, 10, 7, 3, 3, 6, 2, 7, 2, 6, 3, 5, 0, 12, 16, 0, 4, 4, 7, 3, 8, 3, 7, 5, 5, 0, 15, 11, 6, 4, 8, 5, 9, 3, 12, 3, 10, 5, 7, 0, 18

Offset: 0

Author

Emeric Deutsch, Nov 27 2015

Keywords

• nonn
• look
• tabl

Comments

Row n contains n+1 entries (n>=0).

Row sums yield the partition numbers (A000041).

T(n,0) = A007690(n).

T(n,n) = A000009(n).

Sum_{k>=0} k*T(n,k) = A103628(n).

Examples

			T(7,5) = 2 because we have [3,2,1,1] and [5,1,1].
Triangle starts:
1;
0,1;
1,0,1;
1,0,0,2;
2,0,1,0,2;
		

Links

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

Crossrefs

Cf. A000009, A000041, A007690, A103628.

Programs

Maple

g := product(1+t^j*x^j+x^(2*j)/(1-x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 25)): 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, k), k = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(expand(b(n-i*j, i-1)*
      `if`(j=1, 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..15);  # Alois P. Heinz, Nov 27 2015

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Expand[b[n-i*j, i-1]*If[j == 1, 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, 15}] // Flatten (* Jean-François Alcover, Jan 29 2016, after Alois P. Heinz *)

Formula

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