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A275441
Sum of the asymmetry degrees of all compositions of n into odd parts.
2
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
0,5
COMMENTS
The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the asymmetry degree of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).
A sequence is palindromic if and only if its asymmetry degree is 0.
REFERENCES
S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
LINKS
V. E. Hoggatt, Jr., and M. Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
FORMULA
G.f.: g(z)= 2z^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]<a[2]<a[3]<..., denoting F(z) = Sum(z^{a[j]},j>=1}, 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).
EXAMPLE
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.
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);
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 *)
PROG
(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
CROSSREFS
Cf. A275440.
Sequence in context: A276425 A129383 A052957 * A197465 A074933 A157253
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Aug 16 2016
STATUS
approved