A276065 Sum of the asymmetry degrees of all compositions of n with parts in {1,5}.
0, 0, 0, 0, 0, 0, 2, 2, 4, 4, 6, 8, 16, 20, 34, 40, 64, 80, 130, 164, 256, 320, 490, 620, 944, 1200, 1800, 2290, 3400, 4344, 6406, 8206, 12008, 15408, 22404, 28810, 41672, 53680, 77258, 99662, 142808, 184480, 263320, 340578, 484392, 627200, 889160, 1152480
Offset: 0
Examples
a(8) = 4 because the compositions of 8 with parts in {1,5} are 5111, 1511, 1151, 1115, and 11111111, and the sum of their asymmetry degrees is 1+1+1+1+0.
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
- Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239.
- 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 (1,1,-1,0,1,2,-3,0,0,1,-3,0,0,0,-1).
Crossrefs
Cf. A276064.
Programs
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Maple
g := 2*z^6/((1-z+z^2)^2*(1-z^2-z^3)^2*(1+z+z^2)*(1-z^2+z^3)): 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_, _} /; Nor[a == 1, a == 5]]], 1]]], {n, 0, 42}] // Flatten (* Michael De Vlieger, Aug 22 2016 *) LinearRecurrence[{1,1,-1,0,1,2,-3,0,0,1,-3,0,0,0,-1},{0,0,0,0,0,0,2,2,4,4,6,8,16,20,34},50] (* Harvey P. Dale, Aug 29 2021 *) -
PARI
concat(vector(6), Vec(2*x^6/((1-x+x^2)^2*(1-x^2-x^3)^2*(1+x+x^2)*(1-x^2+x^3)) + O(x^50))) \\ Colin Barker, Aug 28 2016
Formula
G.f.: g(z) = 2*z^6/((1-z+z^2)^2*(1-z^2-z^3)^2*(1+z+z^2)*(1-z^2+z^3)). 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*A276064(n,k).
Comments