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

0,7

Comments

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

Links

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

Example

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;

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 *)

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.

Sequence in context: A088705 A065712 A153172 * A321927 A016194 A258139

Adjacent sequences: A242495 A242496 A242497 * A242499 A242500 A242501

Keyword

nonn,tabf

Author

Alois P. Heinz, May 16 2014

Status

approved