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A276059
Sum of the asymmetry degrees of all compositions of n with parts in {3,4,5,6, ...}.
2
0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 6, 10, 14, 24, 38, 62, 98, 156, 242, 376, 580, 896, 1380, 2126, 3266, 5008, 7658, 11688, 17804, 27084, 41148, 62448, 94668, 143360, 216864, 327726, 494790, 746368, 1124950, 1694286, 2549942, 3835120, 5764274, 8658442, 12997998, 19501468
OFFSET
0,8
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^7*(1-z)/((1+z)(1-z+z^3)(1-z-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*A276058(n,k), k>=0).
EXAMPLE
a(7) = 2 because the compositions of 7 with parts in {3,4,5,...} are 7, 34, and 43 and the sum of their asymmetry degrees is 0+1+1.
MAPLE
g := 2*z^7*(1-z)/((1+z)*(1-z+z^3)*(1-z-z^3)^2): gser := series(g, z = 0, 55): seq(coeff(gser, z, n), n = 0 .. 50);
MATHEMATICA
CoefficientList[Series[2 x^7*(1 - x)/((1 + x) (1 - x + x^3) (1 - x - x^3)^2), {x, 0, 45}], x] (* Michael De Vlieger, Aug 28 2016 *)
LinearRecurrence[{2, 0, -1, 0, -1, 2, 0, -1, -1, -1}, {0, 0, 0, 0, 0, 0, 0, 2, 2, 4}, 50] (* Harvey P. Dale, Sep 11 2019 *)
PROG
(PARI) concat(vector(7), Vec(2*x^7*(1-x)/((1+x)*(1-x+x^3)*(1-x-x^3)^2) + O(x^20))) \\ Colin Barker, Aug 28 2016
CROSSREFS
Cf. A276058.
Sequence in context: A300415 A355908 A306731 * A034410 A192682 A050194
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Aug 22 2016
STATUS
approved