|
%I #22 Jan 17 2017 11:22:24
%S 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,
%T 82,99,25,6,14,74,141,230,41,12,202,260,451,85,26,6,22,351,514,953,
%U 148,54,24,766,1049,1798,355,104,18,104,1301,2321,3503,751,194
%N 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.
%C T(n^2,n) = T(n^2+n,-n) = n! = A000142(n) for n>=0.
%H Alois P. Heinz, <a href="/A242626/b242626.txt">Rows n = 0..500, flattened</a>
%e 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].
%e Triangle T(n,k) begins:
%e : n\k : -3 -2 -1 0 1 2 3 ...
%e +-----+------------------------------------
%e : 0 : 1;
%e : 1 : 1;
%e : 2 : 1, 0, 1;
%e : 3 : 2, 2;
%e : 4 : 2, 3, 1, 2;
%e : 5 : 11, 2, 3;
%e : 6 : 2, 2, 14, 8, 6;
%e : 7 : 6, 33, 14, 11;
%e : 8 : 5, 15, 43, 45, 20;
%e : 9 : 44, 82, 99, 25, 6;
%e : 10 : 14, 74, 141, 230, 41, 12;
%e : 11 : 202, 260, 451, 85, 26;
%e : 12 : 6, 22, 351, 514, 953, 148, 54;
%e : 13 : 24, 766, 1049, 1798, 355, 104;
%e : 14 : 18, 104, 1301, 2321, 3503, 751, 194;
%p b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p expand(add(`if`(j=0, 1, x^(2*irem(i, 2)-1))*
%p b(n-i*j, i-1, p+j)/j!, j=0..n/i))))
%p end:
%p T:= n->(p->seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2, 0)):
%p seq(T(n), n=0..20);
%t 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 *)
%Y Columns k=(-5)-5 give: A242836, A242837, A242838, A242839, A242840, A242821, A242841, A242842, A242843, A242844, A242845.
%Y Row sums give A011782.
%Y Cf. A242498 (compositions with multiplicity), A242618 (partitions without multiplicity).
%K nonn,tabf
%O 0,6
%A _Alois P. Heinz_, May 19 2014
|
