A275441 Sum of the asymmetry degrees of all compositions of n into odd parts.
0, 0, 0, 0, 2, 2, 6, 8, 22, 30, 70, 100, 220, 320, 668, 988, 1994, 2982, 5858, 8840, 17010, 25850, 48910, 74760, 139512, 214272, 395256, 609528, 1113362, 1722890, 3120510, 4843400, 8708110, 13551510, 24207958, 37759468, 67068244, 104827712, 185250068
Offset: 0
Examples
a(6) = 6 because the compositions of 6 into odd parts are 15, 51, 33, 1113, 1131, 1311, 3111, 111111 and the sum of their asymmetry degrees is 1 + 1 + 0 + 1 + 1 + 1 + 1 + 0 = 6.
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
- V. E. Hoggatt, Jr. and M. Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
- Index entries for linear recurrences with constant coefficients, signature (1,3,-2,0,-2,-3,1,1).
Crossrefs
Cf. A275440.
Programs
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Maple
g:= 2*z^4*(1-z^2)/((1+z^2)*(1+z-z^2)*(1-z-z^2)^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)/2, Ceiling[Length[#]/2]], Reverse@ Take[(# - 1)/2, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {_, a_, _} /; EvenQ@ a]], 1]]], {n, 0, 30}] // Flatten (* Michael De Vlieger, Aug 17 2016 *) LinearRecurrence[{1,3,-2,0,-2,-3,1,1},{0,0,0,0,2,2,6,8},40] (* Harvey P. Dale, Jan 13 2019 *) -
PARI
concat(vector(4), Vec(2*x^4*(1-x^2)/((1+x^2)*(1+x-x^2)*(1-x-x^2)^2) + O(x^50))) \\ Colin Barker, Aug 29 2016
Formula
G.f.: g(z)= 2*z^4*(1-z^2)/((1+z^2)*(1+z-z^2)*(1-z-z^2)^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*A275440(n,k).
Comments