A275443 Sum of the asymmetry degrees of all compositions of n without 2's.
0, 0, 0, 0, 2, 4, 8, 16, 34, 68, 134, 260, 502, 960, 1824, 3444, 6472, 12108, 22566, 41912, 77608, 143312, 263990, 485196, 889938, 1629256, 2977642, 5433344, 9899776, 18013288, 32734928, 59417944, 107732106, 195130092, 353087560, 638329168, 1153012298
Offset: 0
Examples
a(5) = 4 because the compositions of 5 without 2's are 5, 41, 14, 311, 131, 113, and 11111 and the sum of their asymmetry degrees is 0+1+1+1+0+1+0=4.
References
- S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
- V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
- Index entries for linear recurrences with constant coefficients, signature (3,-2,0,1,-3,1,-1).
Programs
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Maple
g := 2*z^4*(1-z)/((1+z)*(1-2*z+z^2-z^3)^2): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);
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Mathematica
Table[Total@ Map[Total, Map[Map[Boole[# >= 1] &, BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {_, a_, _} /; a == 2]], 1]]], {n, 0, 25}] // Flatten (* Michael De Vlieger, Aug 17 2016 *) -
PARI
concat(vector(4), Vec(2*x^4*(1-x)/((1+x)*(1-2*x+x^2-x^3)^2) + O(x^50))) \\ Colin Barker, Aug 29 2016
Formula
G.f.: g(z) = 2*z^4*(1-z)/((1+z)*(1-2*z+z^2-z^3)^2). In the more general situation of compositions into a[1]=1} z^(a[j]), we have g(z) = (F(z)^2 - F(z^2))/((1+F(z))*(1-F(z))^2).
a(n) = Sum_{k >= 0} k*A275442(n,k).
Comments