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A275445
Sum of the asymmetry degrees of all compositions of n with parts in {1,2,3}.
2
0, 0, 0, 2, 4, 10, 22, 50, 106, 222, 458, 936, 1890, 3788, 7540, 14924, 29388, 57620, 112540, 219062, 425112, 822726, 1588314, 3059470, 5881254, 11284514, 21614774, 41336300, 78936358, 150533496, 286708744, 545428024, 1036468344, 1967555208, 3731449176, 7070218506, 13384916364, 25319020898, 47857031870, 90391975562, 170614347714
OFFSET
0,4
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
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.
FORMULA
G.f. g(z) = 2*z^3*(1+z+z^2)/((1+z)(1+z^2)(1-z-z^2-z^3)^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*A275444(n,k).
4*a(n) = (-1)^(n+1) +A057077(n+1) -2*A000073(n) +4*A073778(n+2). - R. J. Mathar, Jan 13 2023
EXAMPLE
a(4) = 4 because the compositions of 4 with parts in {1,2,3} are 13, 31, 22, 211, 121, 112, and 1111 and the sum of their asymmetry degrees is 1 + 1 + 0 + 1 + 0 + 1 + 0 = 4.
MAPLE
g := 2*z^3*(1+z+z^2)/((1+z)*(1+z^2)*(1-z-z^2-z^3)^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, Ceiling[Length[#]/2]], ReverseTake[# - 1, -Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {a_, ___} /; a > 3]], 1]]], {n, 0, 24}] // Flatten (* Michael De Vlieger, Aug 17 2016 *)
PROG
(PARI) concat(vector(3), Vec(2*x^3*(1+x+x^2)/((1+x)*(1+x^2)*(1-x-x^2-x^3)^2) + O(x^50))) \\ Colin Barker, Aug 28 2016
CROSSREFS
Cf. A275444.
Sequence in context: A239075 A260916 A192627 * A075560 A078040 A240041
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Aug 17 2016
STATUS
approved