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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
OFFSET
0,6
COMMENTS
T(n^2,n) = T(n^2+n,-n) = n! = A000142(n) for n>=0.
LINKS
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
Row sums give A011782.
Cf. A242498 (compositions with multiplicity), A242618 (partitions without multiplicity).
Sequence in context: A376307 A227738 A103960 * A360387 A306399 A376076
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, May 19 2014
STATUS
approved