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

0,6

Comments

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

Links

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

Example

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;

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 *)

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).

Sequence in context: A209254 A227738 A103960 * A360387 A306399 A240689

Adjacent sequences: A242623 A242624 A242625 * A242627 A242628 A242629

Keyword

nonn,tabf

Author

Alois P. Heinz, May 19 2014

Status

approved