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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 (list; graph; refs; listen; history; text; internal format)
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), 233-239.

V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

FORMULA

G.f.: G(t,z) = (1+z+z^3)/(1-z^2-2tz^4-z^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))/(1-F(z^2)-t(F(z)^2-F(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)/(1-z^2-2*t*z^4-z^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: A304777 A058762 A241314 * A276060 A276058 A247302

Adjacent sequences:  A276053 A276054 A276055 * A276057 A276058 A276059

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Aug 18 2016

STATUS

approved

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Last modified October 20 17:20 EDT 2018. Contains 316392 sequences. (Running on oeis4.)