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%I #12 Aug 17 2016 13:10:29
%S 1,1,2,1,2,3,2,2,6,5,4,4,3,14,4,8,10,16,5,30,12,8,13,20,48,8,8,60,36,
%T 40,21,40,124,32,16,13,116,88,144,16,34,76,292,112,96,21,218,204,432,
%U 80,32,55,142,648,320,400,32,34,402,444,1160,320,224
%N Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,2} having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/3)).
%C 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).
%C Number of entries in row n is 1 + floor(n/3).
%C Sum of entries in row n is A000045(n+1) (Fibonacci).
%C T(n,0) = A053602(n+1) (= number of palindromic compositions of n with parts in {1,2}).
%C Sum_{k>=0} k*T(n,k) = A275439(n).
%H Krithnaswami Alladi and V. E. Hoggatt, Jr. <a href="http://www.fq.math.ca/Scanned/13-3/alladi1.pdf">Compositions with Ones and Twos</a>, Fibonacci Quarterly, 13 (1975), 233-239.
%H V. E. Hoggatt, Jr., and Marjorie Bicknell, <a href="http://www.fq.math.ca/Scanned/13-4/hoggatt1.pdf">Palindromic compositions</a>, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
%F G.f.: G(t,z) = (1+z+z^2)/(1-z^2-2tz^3-z^4). 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.
%e Row 4 is [3,2] because the compositions of 4 with parts in {1,2} are 22, 112, 121, 211, and 1111, having asymmetry degrees 0, 1, 0, 1, 0, respectively.
%e Triangle starts:
%e 1;
%e 1;
%e 2;
%e 1,2;
%e 3,2;
%e 2,6;
%e 5,4,4.
%p G:=(1+z+z^2)/(1-z^2-2*t*z^3-z^4): Gser:=simplify(series(G,z=0,25)): for n from 0 to 20 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 20 do seq(coeff(P[n],t,j),j=0..degree(P[n])) end do; # yields sequence in triangular form
%t Join[{{1}}, Table[BinCounts[#, {0, 1 + Floor[n/3], 1}] &@ Map[Total, Map[BitXor[Take[# - 1, Ceiling[Length[#]/2]], Reverse@ Take[# - 1, -Ceiling[Length[#]/2]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {a_, ___} /; a > 2]], 1]]], {n, 17}]] // Flatten (* _Michael De Vlieger_, Aug 17 2016 *)
%Y Cf. A000045, A053602, A275439.
%K nonn,tabf
%O 0,3
%A _Emeric Deutsch_, Aug 16 2016
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