A276063 Sum of the asymmetry degrees of all compositions of n with parts in {1,4}.
0, 0, 0, 0, 0, 2, 2, 4, 4, 8, 14, 20, 32, 44, 70, 104, 152, 228, 326, 488, 704, 1026, 1492, 2144, 3120, 4470, 6450, 9256, 13256, 19026, 27144, 38840, 55360, 78910, 112406, 159768, 227240, 322500, 457734, 648996, 919372, 1302114, 1842036, 2605452, 3682112
Offset: 0
Examples
a(6) = 2 because the compositions of 6 with parts in {1,4} are 411, 141, 114, and 111111 and the sum of their asymmetry degrees is 1+0+1+0.
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,-1,1,2,-3,0,1,-3,0,0,-1).
Crossrefs
Cf. A276062.
Programs
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Maple
g := 2*z^5/((1+z+z^4)*(1-z-z^4)^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 == 4]]], 1]]], {n, 0, 38}] // Flatten (* Michael De Vlieger, Aug 22 2016 *) -
PARI
concat(vector(5), Vec(2*x^5/((1+x+x^4)*(1-x-x^4)^2) + O(x^50))) \\ Colin Barker, Aug 28 2016
Formula
G.f.: g(z) = 2*z^5/((1+z+z^4)*(1-z-z^4)^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*A276062(n,k).
Comments