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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 (list; table; graph; refs; listen; history; text; internal format)
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

Alois P. Heinz, Rows n = 0..140, flattened

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

Adjacent sequences:  A239363 A239364 A239365 * A239367 A239368 A239369

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Mar 20 2014

STATUS

approved

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Last modified October 22 10:16 EDT 2021. Contains 348160 sequences. (Running on oeis4.)