A242498 Number T(n,k) of compositions of n, where k is the difference between the number of odd parts and the number of even parts; triangle T(n,k), n>=0, -floor(n/2)+(n mod 2)<=k<=n, read by rows.

1, 1, 1, 0, 0, 1, 2, 1, 0, 1, 1, 1, 0, 3, 2, 0, 1, 3, 4, 1, 4, 3, 0, 1, 1, 2, 1, 6, 9, 3, 5, 4, 0, 1, 4, 9, 6, 11, 16, 6, 6, 5, 0, 1, 1, 3, 3, 11, 24, 18, 19, 25, 10, 7, 6, 0, 1, 5, 16, 18, 28, 51, 40, 31, 36, 15, 8, 7, 0, 1, 1, 4, 6, 19, 51, 60, 65, 95, 75, 48, 49, 21, 9, 8, 0, 1

Offset: 0

Author

Alois P. Heinz, May 16 2014

Keywords

• nonn
• tabf

Comments

T(n,k) = T(n+k,-k).

Examples

			Triangle T(n,k) begins:
: n\k : -5 -4 -3  -2  -1   0   1   2   3   4   5   6  7  8  9 10 ...
+-----+---------------------------------------------------------
:  0  :                    1;
:  1  :                        1;
:  2  :                1,  0,  0,  1;
:  3  :                    2,  1,  0,  1;
:  4  :            1,  1,  0,  3,  2,  0,  1;
:  5  :                3,  4,  1,  4,  3,  0,  1;
:  6  :        1,  2,  1,  6,  9,  3,  5,  4,  0,  1;
:  7  :            4,  9,  6, 11, 16,  6,  6,  5,  0, 1;
:  8  :     1, 3,  3, 11, 24, 18, 19, 25, 10,  7,  6, 0, 1;
:  9  :        5, 16, 18, 28, 51, 40, 31, 36, 15,  8, 7, 0, 1;
: 10  :  1, 4, 6, 19, 51, 60, 65, 95, 75, 48, 49, 21, 9, 8, 0, 1;
		

Links

• Alois P. Heinz, Rows n = 0..120, flattened

Crossrefs

Columns k=0-10 gives: A098123, A242499, A242500, A242501, A242502, A242503, A242504, A242505, A242506, A242507, A242508.

Row sums give A011782.

Diagonals include: A000012, A000004, A001477, A000217, A000290, A180415.

Row lengths give A016777(floor(n/2)).

Cf. A240009, A240021.

Programs

Maple

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

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i<1, 0, Expand[Sum[x^(j*(2*Mod[i, 2]-1))*b[n-i*j, i-1, p+j]/j!, {j, 0, n/i}]]]] ; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, Exponent[p, x, Min], Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)