A242618 Number T(n,k) of partitions of n, where k is the difference between the number of odd parts and the number of even parts, both counted without multiplicity; triangle T(n,k), n>=0, read by rows.

1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 1, 4, 2, 1, 1, 2, 3, 3, 2, 1, 8, 3, 3, 2, 4, 6, 5, 5, 4, 13, 8, 4, 1, 5, 5, 11, 13, 7, 1, 11, 20, 14, 9, 2, 1, 6, 13, 17, 26, 11, 3, 1, 22, 31, 27, 15, 5, 2, 12, 18, 34, 44, 18, 7, 4, 40, 47, 51, 23, 11, 5, 16, 36, 56, 72, 34, 11, 1

Offset: 0

Author

Alois P. Heinz, May 19 2014

Keywords

• nonn
• tabf
• look

Comments

T(n,0) = A241638(n).

Sum_{k<0} T(n,k) = A241640(n).

Sum_{k<=0} T(n,k) = A241639(n).

Sum_{k>=0} T(n,k) = A241637(n).

Sum_{k>0} T(n,k) = A241636(n).

T(n^2,n) = T(n^2+n,-n) = 1.

T(n^2+n,n) = Sum_{k} T(n,k) = A000041(n).

T(n^2+3*n,-n) = A000712(n).

Examples

			Triangle T(n,k) begins:
: n\k : -3  -2  -1   0   1   2   3 ...
+-----+---------------------------
:  0  :              1;
:  1  :                  1;
:  2  :          1,  0,  1;
:  3  :              1,  2;
:  4  :          2,  1,  1,  1;
:  5  :              4,  2,  1;
:  6  :      1,  2,  3,  3,  2;
:  7  :          1,  8,  3,  3;
:  8  :      2,  4,  6,  5,  5;
:  9  :          4, 13,  8,  4,  1;
: 10  :      5,  5, 11, 13,  7,  1;
: 11  :         11, 20, 14,  9,  2;
: 12  :  1,  6, 13, 17, 26, 11,  3;
: 13  :      1, 22, 31, 27, 15,  5;
: 14  :  2, 12, 18, 34, 44, 18,  7;
		

Links

• Alois P. Heinz, Rows n = 0..500, flattened

Crossrefs

Columns k=(-10)-10 give: A242682, A242683, A242684, A242685, A242686, A242687, A242688, A242689, A242690, A242691, A241638, A242692, A242693, A242694, A242695, A242696, A242697, A242698, A242699, A242700, A242701.

Row sums give A000041.

Cf. A240009 (parts counted with multiplicity), A240021 (distinct parts), A242626 (compositions counted without multiplicity).

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(b(n, i-1)+add(b(n-i*j, i-1)*x^(2*irem(i, 2)-1), j=1..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2)):
seq(T(n), n=0..20);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Expand[b[n, i - 1] + Sum[b[n - i*j, i - 1]*x^(2*Mod[i, 2] - 1), {j, 1, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, Exponent[p, x, Min], Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 12 2016 after Alois P. Heinz *)