A276057 Sum of the asymmetry degrees of all compositions of n with parts in {1,3}.
0, 0, 0, 0, 2, 2, 4, 6, 14, 18, 36, 50, 94, 130, 236, 330, 580, 816, 1404, 1984, 3354, 4758, 7932, 11286, 18600, 26532, 43308, 61908, 100232, 143540, 230776, 331008, 528950, 759726, 1207584, 1736534, 2747242, 3954826, 6230444, 8977686, 14090410, 20320854
Offset: 0
Examples
a(6) = 4 because the compositions of 6 with parts in {1,3} are 33, 3111, 1311, 1131, 1113, and 111111 and the sum of their asymmetry degrees is 0 + 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 (1,1,0,2,-3,1,-3,0,-1).
Crossrefs
Cf. A276056.
Programs
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Maple
g:=2*z^4/((1+z+z^3)*(1-z-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_, _} /; Nor[a == 1, a == 3]]], 1]]], {n, 0, 34}] // Flatten (* or *) CoefficientList[Series[2 x^4/((1 + x + x^3) (1 - x - x^3)^2), {x, 0, 41}], x] (* Michael De Vlieger, Aug 28 2016 *) -
PARI
concat(vector(4), Vec(2*x^4/((1+x+x^3)*(1-x-x^3)^2) + O(x^50))) \\ Colin Barker, Aug 28 2016
Formula
G.f.: g(z) = 2*z^4/((1+z+z^3)*(1-z-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*A276056(n,k).
Comments