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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.
1
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, 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, 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
OFFSET
0,11
COMMENTS
Row sums = 2^floor(n/2).
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
LINKS
FORMULA
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).
EXAMPLE
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, 0, 1,
5, 0, 5, 0, 4, 0, 1, 0, 1,
3, 2, 3, 2, 1, 3, 1, 0, 0, 1
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.
MAPLE
b:= proc(n) option remember; `if`(n=0, 1, expand(
add(b(n-j)*`if`(j=1, x^2, 1), j=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))
(add(b(i)*`if`(n-2*i=1, x, 1), i=0..n/2)):
seq(T(n), n=0..30); # Alois P. Heinz, Mar 21 2014
MATHEMATICA
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
CROSSREFS
Sequence in context: A050374 A238877 A047886 * A014944 A015879 A230000
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Mar 20 2014
STATUS
approved