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 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. 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row sums = A239632(n). LINKS P. Z. Chinn, R. Grimaldi, and S. Heubach, The Frequency of Summands of a Particular Size in Palindromic Compositions, Ars Combinatoria 69 (2003), 65-78. FORMULA Explicit formula for T(n,k) given in reference [Chinn, Grimaldi, Heubach] as Theorem 6: a(n) = 0 if n=2k and n!=k (mod 2); a(n) = 1 if n=k; a(n) = 2^((n-k)/2-1) if k=2k and n==k (mod 2). (end) 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, A078836, A078836, A079861, A079862, A079863. 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

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Last modified September 15 12:22 EDT 2019. Contains 327078 sequences. (Running on oeis4.)