cp's OEIS Frontend

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.

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