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

0,7

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).

Links

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

Example

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;

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 *)

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).

Sequence in context: A104320 A350818 A340142 * A180264 A225200 A128706

Adjacent sequences: A242615 A242616 A242617 * A242619 A242620 A242621

Keyword

nonn,tabf,look

Author

Alois P. Heinz, May 19 2014

Status

approved