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A239366 Triangular array read by rows: T(n,k) is the number of palindromic compositions of n having exactly k 1's, n>=0, 0<=k<=n.

%I A239366 #24 Mar 21 2014 20:05:06
%S A239366 1,0,1,1,0,1,1,0,0,1,2,0,1,0,1,1,1,1,0,0,1,3,0,3,0,1,0,1,2,1,1,2,1,0,
%T A239366 0,1,5,0,5,0,4,0,1,0,1,3,2,3,2,1,3,1,0,0,1,8,0,10,0,7,0,5,0,1,0,1,5,3,
%U A239366 5,5,4,3,1,4,1,0,0,1,13,0,18,0,16,0,9,0,6,0,1,0,1,8,5,10,8,7,9,5,4,1,5,1,0,0,1
%N A239366 Triangular array read by rows: T(n,k) is the number of palindromic compositions of n having exactly k 1's, n>=0, 0<=k<=n.
%C A239366 Row sums = 2^floor(n/2).
%C A239366 T(n,0) = A053602(n-1) for n>0, T(n,1) = A079977(n-5) for n>4, T(2n+1,3) = A006367(n-1) for n>0, both bisections of column k=2 contain A010049. - _Alois P. Heinz_, Mar 21 2014
%H A239366 Alois P. Heinz, <a href="/A239366/b239366.txt">Rows n = 0..140, flattened</a>
%F A239366 G.f.: G(x,y) = ((1 + x)*(1 - x + x^2 + x*y - x^2*y))/(1 - x^2 - x^4 - x^2*y^2 + x^4*y^2). Satisfies G(x,y) = 1/(1 - x) - x + y*x + (x^2/(1 - x^2) - x^2 +y^2*x^2)*G(x,y).
%e A239366 1,
%e A239366 0, 1,
%e A239366 1, 0, 1,
%e A239366 1, 0, 0, 1,
%e A239366 2, 0, 1, 0, 1,
%e A239366 1, 1, 1, 0, 0, 1,
%e A239366 3, 0, 3, 0, 1, 0, 1,
%e A239366 2, 1, 1, 2, 1, 0, 0, 1,
%e A239366 5, 0, 5, 0, 4, 0, 1, 0, 1,
%e A239366 3, 2, 3, 2, 1, 3, 1, 0, 0, 1
%e A239366 There are eight palindromic compositions of 6: T(6,0)=3 because we have: 6, 3+3, 2+2+2.  T(6,2)=3 because we have: 1+4+1, 2+1+1+2, 1+2+2+1.  T(6,4)=1 because we have: 1+1+2+1+1. T(6,6)=1 because we have: 1+1+1+1+1+1.
%p A239366 b:= proc(n) option remember;  `if`(n=0, 1, expand(
%p A239366       add(b(n-j)*`if`(j=1, x^2, 1), j=1..n)))
%p A239366     end:
%p A239366 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))
%p A239366     (add(b(i)*`if`(n-2*i=1, x, 1), i=0..n/2)):
%p A239366 seq(T(n), n=0..30);  # _Alois P. Heinz_, Mar 21 2014
%t A239366 nn=15;Table[Take[CoefficientList[Series[((1+x)*(1-x+x^2+x*y-x^2*y))/(1-x^2-x^4-x^2*y^2+x^4*y^2),{x,0,nn}],{x,y}][[n]],n],{n,1,nn}]//Grid
%K A239366 nonn,tabl
%O A239366 0,11
%A A239366 _Geoffrey Critzer_, Mar 20 2014