A239631 Triangular array read by rows: T(n,k) is the number of parts equal to k over all palindromic compositions of n, n>=1, 1<=k<=n.

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, 224, 112, 40, 24, 6, 6, 0, 2, 0, 1, 0, 1

Offset

1,2

Links

Table of n, a(n) for n=1..78.

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

Cf. A016116, A079859, A078836, A079861, A079862, A079863, A239632 (row sums).

Sequence in context: A137587 A168021 A137639 * A113288 A199580 A035215

Adjacent sequences: A239628 A239629 A239630 * A239632 A239633 A239634

Keyword

nonn,tabl

Author

Geoffrey Critzer, Mar 22 2014

Status

approved