OFFSET
1,2
LINKS
Phyllis Zweig Chinn, Ralph Grimaldi and Silvia Heubach, The Frequency of Summands of a Particular Size in Palindromic Compositions, Ars Combinatoria 69 (2003), 65-78.
FORMULA
Explicit formulas for T(n,k) given in reference [Chinn, Grimaldi, Heubach] as Theorem 6:
T(n,k) = 0 if n<k or if k<n<2k and n!=k (mod 2);
T(n,k) = 2^(floor(n/2)-k)*(2 + floor(n/2) - k) if n>=2k and n!=k (mod 2);
T(n,k) = 1 if n=k;
T(n,k) = 2^((n-k)/2-1) if k<n<2k and n==k (mod 2);
T(n,k) = 2^(floor(n/2)-k)*(2 + floor(n/2) - k + 2^floor((k+1)/2-1)) if n>=2k and n==k (mod 2).
O.g.f. for column k: x^k/(1-F(x^2)) + 2*x^(2*k)*(1 + F(x))/(1 - F(x^2))^2 where F(x)= x/(1-x).
EXAMPLE
1,
2, 1,
3, 0, 1,
6, 3, 0, 1,
8, 2, 1, 0, 1,
16, 8, 2, 1, 0, 1,
20, 6, 4, 0, 1, 0, 1,
40, 20, 6, 4, 0, 1, 0, 1,
48, 16, 10, 2, 2, 0, 1, 0, 1,
96, 48, 16, 10,2, 2, 0, 1, 0, 1,
112, 40, 24, 6, 6, 0, 2, 0, 1, 0, 1
In the palindromic compositions of 5: 5, 1+3+1, 2+1+2, 1+1+1+1+1 there are T(5,1)=8 ones, T(5,2)=2 twos, and T(5,3)=1 three and T(5,5)=1 five.
MATHEMATICA
nn=15; Table[Take[Drop[Transpose[Map[PadRight[#, nn+1]&, Level[Table[r=Solve[p==1/(1-x)-x^n+y x^n+(x^2/(1-x^2)-x^(2n)+y^2x^(2n))p, p]; CoefficientList[Series[D[p/.r, y]/.y->1, {x, 0, nn}], x], {n, 1, nn}], {2}]]], 1][[n]], n], {n, 1, nn}]//Grid
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Mar 22 2014
STATUS
approved