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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 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]=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 January 20 03:32 EST 2019. Contains 319323 sequences. (Running on oeis4.)