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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. 13
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 * A240689 A233567 A141059

Adjacent sequences:  A242623 A242624 A242625 * A242627 A242628 A242629

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, May 19 2014

STATUS

approved

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Last modified August 19 08:17 EDT 2019. Contains 326115 sequences. (Running on oeis4.)