A264404 Triangle read by rows: T(n,k) is the number of partitions of n in which the sum of the parts of multiplicity greater than 1 is k (0<=k). For example, in the partition [3,2,2,1,1,1] the sum k is 2 + 1 = 3.

1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 3, 1, 4, 4, 1, 2, 5, 6, 2, 2, 6, 8, 3, 3, 2, 8, 11, 4, 5, 2, 10, 14, 5, 7, 3, 3, 12, 19, 7, 10, 5, 3, 15, 24, 9, 15, 6, 4, 4, 18, 31, 12, 20, 9, 7, 4, 22, 39, 15, 26, 13, 9, 6, 5, 27, 49, 19, 36, 17, 13, 10, 5, 32, 61, 24, 46, 23, 18, 14, 7, 6, 38, 76, 30, 60, 31, 24

Offset: 0

Author

Emeric Deutsch, Nov 27 2015

Keywords

• nonn
• tabf

Comments

Only one copy of each part of multiplicity greater than one is used.

Row n contains floor(n/2) entries (n>=0).

Row sums yield the partition numbers (A000041).

T(n,0) = A000009(n).

Sum_{k>=0} k*T(n,k) = A103650(n).

Examples

			T(9,3) = 5 because we have [3,3,3], [3,3,2,1], [3,2,2,1,1], [2,2,2,1,1,1], and [2,2,1,1,1,1,1].
Triangle starts:
1;
1;
1,1;
2,1;
2,2,1;
3,3,1;
4,4,1,2;
		

Links

• Alois P. Heinz, Rows n = 0..350, flattened

Crossrefs

Cf. A000009, A000041, A103650.

Programs

Maple

g := product(1+x^j+t^j*x^(2*j)/(1-x^j), j = 1 .. 100): gser := simplify(series(g, x = 0, 35)): 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, k), k = 0 .. floor((1/2)*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)*x^`if`(j>1, i, 0)), 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..20);  # Alois P. Heinz, Nov 29 2015

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Expand[b[n - i*j, i - 1]*x^If[j > 1, i, 0]], {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, 20}] // Flatten (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

Formula

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