

A275446


Triangle read by rows: T(n,k) is the number of compositions of n with parts in {2,1,3,5,7,9,...} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/3)).


3



1, 1, 2, 2, 2, 3, 4, 4, 10, 6, 16, 4, 8, 30, 12, 11, 48, 36, 15, 82, 76, 8, 21, 128, 164, 32, 29, 204, 312, 112, 40, 312, 596, 288, 16, 55, 482, 1064, 704, 80, 76, 728, 1884, 1536, 320, 105, 1100, 3212, 3248, 960, 32, 145, 1640, 5428, 6464, 2624, 192
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OFFSET

0,3


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/3).
Sum of entries in row n is A052535(n).
T(n,0) = A103632(n+2).
Sum_{k>=0} k*T(n,k) = A275447(n).


REFERENCES

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


LINKS

Table of n, a(n) for n=0..56.
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) = (1z^4)/(1zz^2+(12t)z^3z^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.


EXAMPLE

Row 4 is [3,4] because the compositions of 4 with parts in {2,1,3,5,7,...} are 22, 31, 13, 211, 121, 112, and 1111, having asymmetry degrees 0, 1, 1, 1, 0, 1, and 0, respectively.
Triangle starts:
1;
1;
2;
2,2;
3,4;
4,10;
6,16,4.


MAPLE

G := (1z^4)/(1zz^2+(12*t)*z^3z^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[BinCounts[#, {0, 1 + Floor[n/3], 1}] &@ 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_, ___} /; And[EvenQ@ a, a != 2]]], 1]]], {n, 0, 16}] // Flatten (* Michael De Vlieger, Aug 17 2016 *)


CROSSREFS

Cf. A052535, A103632, A275447.
Sequence in context: A015750 A275442 A303846 * A306271 A144732 A055224
Adjacent sequences: A275443 A275444 A275445 * A275447 A275448 A275449


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Aug 17 2016


STATUS

approved



