

A276056


Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,3} and having asymmetry degree equal to k, (n>=0; 0<=k<=floor(n/4)).


1



1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 3, 6, 3, 6, 4, 5, 10, 4, 4, 12, 12, 7, 18, 16, 6, 22, 24, 8, 10, 34, 36, 8, 9, 36, 52, 32, 15, 58, 76, 40, 13, 60, 108, 80, 16, 22, 96, 160, 112, 16, 19, 100, 204, 192, 80, 32, 160, 312, 272, 96, 28, 162, 376, 440, 240, 32
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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).
Number of entries in row n is 1 + floor(n/4).
Sum of entries in row n is A000930(n).


REFERENCES

S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.


LINKS

Table of n, a(n) for n=0..65.
Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233239.
V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350356.


FORMULA

G.f.: G(t,z) = (1+z+z^3)/(1z^22tz^4z^6). 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(t,z) =(1+F(z))/(1F(z^2)t(F(z)^2F(z^2))). In particular, for t=0 we obtain Theorem 1.2 of the Hoggatt et al. reference.
T(n,0) = A226916(n+7).
Sum(k*T(n,k), k>=0) = A276057(n).


EXAMPLE

Row 6 is [2,4] because the compositions of 6 with parts in {1,3} are 33, 3111, 1311, 1131, 1113, and 111111, having asymmetry degrees 0, 1, 1, 1, 1, and 0, respectively.
Triangle starts:
1;
1;
1;
2;
1, 2;
2, 2;
2, 4;
...


MAPLE

G := (1+z+z^3)/(1z^22*t*z^4z^6): Gser := simplify(series(G, z = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form


MATHEMATICA

Table[TakeWhile[BinCounts[#, {0, 1 + Floor[n/4], 1}], # != 0 &] &@ 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 == 3]]], 1]]], {n, 0, 20}] // Flatten (* Michael De Vlieger, Aug 28 2016 *)


CROSSREFS

Cf. A000930, A226916, A276057.
Sequence in context: A277210 A058762 A241314 * A276060 A276058 A247302
Adjacent sequences: A276053 A276054 A276055 * A276057 A276058 A276059


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Aug 18 2016


STATUS

approved



