A276426 Triangle read by rows: T(n,k) is the number of integer partitions of n having k distinct odd parts (n>=0).

1, 0, 1, 1, 1, 0, 3, 2, 2, 1, 0, 6, 1, 3, 5, 3, 0, 11, 4, 5, 8, 9, 0, 20, 9, 1, 7, 15, 19, 1, 0, 32, 21, 3, 11, 24, 38, 4, 0, 51, 41, 9, 15, 39, 69, 12, 0, 80, 73, 23, 22, 58, 123, 27, 1, 0, 119, 128, 49, 1, 30, 90, 202, 60, 3, 0, 175, 213, 98, 4, 42, 130, 328, 118, 9

Offset: 0

Author

Emeric Deutsch, Sep 19 2016

Keywords

• nonn
• tabf

Comments

Sum of entries in row n = A000041(n).

T(2n,0) = A000041(n); T(2n+1,0) = 0.

Sum(k*T(n,k), k>=0) = A024786(n+1).

Examples

			T(4,0) = 2 because we have [4], [2,2];
T(4,1) = 2 because we have [1,1,2], [1,1,1,1];
T(4,2) = 1 because we have [1,3];
Triangle starts:
1;
0,1;
1,1;
0,3;
2,2,1.
		

Links

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

Crossrefs

Cf. A000041, A024786, A264052.

Programs

Maple

G := product((1-x^(2*j-1)+t*x^(2*j-1))/(1-x^j), j = 1 .. 100): Gser := simplify(series(G, x = 0, 32)); for n from 0 to 27 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 27 do seq(coeff(P[n], t, i), i = 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>0 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..25);  # Alois P. Heinz, Sep 20 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 > 0 && 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, 25}] // Flatten (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

Formula

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