A242626 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, both counted without multiplicity; triangle T(n,k), n>=0, read by rows.

1, 1, 1, 0, 1, 2, 2, 2, 3, 1, 2, 11, 2, 3, 2, 2, 14, 8, 6, 6, 33, 14, 11, 5, 15, 43, 45, 20, 44, 82, 99, 25, 6, 14, 74, 141, 230, 41, 12, 202, 260, 451, 85, 26, 6, 22, 351, 514, 953, 148, 54, 24, 766, 1049, 1798, 355, 104, 18, 104, 1301, 2321, 3503, 751, 194

Offset: 0

Author

Alois P. Heinz, May 19 2014

Keywords

• nonn
• tabf

Comments

T(n^2,n) = T(n^2+n,-n) = n! = A000142(n) for n>=0.

Examples

			T(8,-1) = 15: [2,2,2,2], [1,1,2,4], [1,1,4,2], [1,2,1,4], [1,2,4,1], [1,4,1,2], [1,4,2,1], [2,1,1,4], [2,1,4,1], [2,4,1,1], [4,1,1,2], [4,1,2,1], [4,2,1,1], [4,4], [8].
Triangle T(n,k) begins:
: n\k : -3   -2    -1     0     1    2    3 ...
+-----+------------------------------------
:  0  :                   1;
:  1  :                         1;
:  2  :             1,    0,    1;
:  3  :                   2,    2;
:  4  :             2,    3,    1,   2;
:  5  :                  11,    2,   3;
:  6  :       2,    2,   14,    8,   6;
:  7  :             6,   33,   14,  11;
:  8  :       5,   15,   43,   45,  20;
:  9  :            44,   82,   99,  25,   6;
: 10  :      14,   74,  141,  230,  41,  12;
: 11  :           202,  260,  451,  85,  26;
: 12  :  6,  22,  351,  514,  953, 148,  54;
: 13  :      24,  766, 1049, 1798, 355, 104;
: 14  : 18, 104, 1301, 2321, 3503, 751, 194;
		

Links

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

Crossrefs

Columns k=(-5)-5 give: A242836, A242837, A242838, A242839, A242840, A242821, A242841, A242842, A242843, A242844, A242845.

Row sums give A011782.

Cf. A242498 (compositions with multiplicity), A242618 (partitions without multiplicity).

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      expand(add(`if`(j=0, 1, x^(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[If[j==0, 1, x^(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, Jan 17 2017, translated from Maple *)