cp's OEIS Frontend

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.

%I A239631 #33 Apr 25 2024 11:37:26
%S A239631 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,
%T A239631 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,
%U A239631 2,0,1,0,1,224,112,40,24,6,6,0,2,0,1,0,1
%N 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.
%H A239631 Phyllis Zweig Chinn, Ralph Grimaldi and Silvia Heubach, <a href="https://www.calstatela.edu/sites/default/files/freqs.pdf">The Frequency of Summands of a Particular Size in Palindromic Compositions</a>, Ars Combinatoria 69 (2003), 65-78.
%F A239631 Explicit formulas for T(n,k) given in reference [Chinn, Grimaldi, Heubach] as Theorem 6:
%F A239631 T(n,k) = 0 if n<k or if k<n<2k and n!=k (mod 2);
%F A239631 T(n,k) = 2^(floor(n/2)-k)*(2 + floor(n/2) - k) if n>=2k and n!=k (mod 2);
%F A239631 T(n,k) = 1 if n=k;
%F A239631 T(n,k) = 2^((n-k)/2-1) if k<n<2k and n==k (mod 2);
%F A239631 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).
%F A239631 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).
%e A239631 1,
%e A239631 2,   1,
%e A239631 3,   0,  1,
%e A239631 6,   3,  0,  1,
%e A239631 8,   2,  1,  0, 1,
%e A239631 16,  8,  2,  1, 0, 1,
%e A239631 20,  6,  4,  0, 1, 0, 1,
%e A239631 40,  20, 6,  4, 0, 1, 0, 1,
%e A239631 48,  16, 10, 2, 2, 0, 1, 0, 1,
%e A239631 96,  48, 16, 10,2, 2, 0, 1, 0, 1,
%e A239631 112, 40, 24, 6, 6, 0, 2, 0, 1, 0, 1
%e A239631 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.
%t A239631 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
%Y A239631 Cf. A016116, A079859, A078836, A079861, A079862, A079863, A239632 (row sums).
%K A239631 nonn,tabl
%O A239631 1,2
%A A239631 _Geoffrey Critzer_, Mar 22 2014