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A275442 Triangle read by rows: T(n,k) is the number of compositions without 2's and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/4)).

%I A275442 #24 May 09 2025 23:13:24
%S A275442 1,1,1,2,2,2,3,4,4,8,5,16,7,26,4,9,44,12,12,70,32,16,108,76,21,166,
%T A275442 156,8,28,248,308,32,37,368,572,104,49,540,1020,288,65,784,1768,696,
%U A275442 16,86,1132,2976,1568,80,114,1622,4908,3304,304,151,2312,7944,6624,960,200,3280,12652,12768,2640,32
%N A275442 Triangle read by rows: T(n,k) is the number of compositions without 2's and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/4)).
%C A275442 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 A275442 Number of entries in row n is 1 + floor(n/4).
%C A275442 Sum of entries in row n is A005251(n+1).
%C A275442 T(n,0) = A000931(n+5) (= number of palindromic compositions of n without 2's).
%C A275442 Sum_{k >= 0} k*T(n,k) = A275443(n).
%D A275442 S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%H A275442 John Tyler Rascoe, <a href="/A275442/b275442.txt">Rows n = 0..200, flattened</a>
%H A275442 Krithnaswami Alladi and V. E. Hoggatt, Jr. <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/13-3/alladi1.pdf">Compositions with Ones and Twos</a>, Fibonacci Quarterly, 13 (1975), 233-239.
%H A275442 P. Chinn and S. Heubach, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Heubach/heubach5.html">Integer Sequences Related to Compositions without 2's</a>, J. Integer Seqs., Vol. 6, 2003.
%H A275442 V. E. Hoggatt, Jr. and Marjorie Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/13-4/hoggatt1.pdf">Palindromic compositions</a>, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
%F A275442 G.f.: G(t,z) = (1-z^2)/(1-z-z^2+z^4-2*t*z^4). In the more general situation of compositions into a[1]<a[2]<a[3]<..., denoting F(z) = Sum_{j>=1} z^(a[j]), 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 A275442 Row 5 is [3,4] because the compositions of 5 without 2's are 5, 113, 131, 311, 14, 41, and 11111, having asymmetry degrees 0, 1, 0, 1, 1, 1, and 0, respectively.
%e A275442 Triangle starts:
%e A275442   1;
%e A275442   1;
%e A275442   1;
%e A275442   2;
%e A275442   2,2;
%e A275442   3,4;
%e A275442   4,8;
%e A275442   5,16.
%p A275442 G := (1-z^2)/(1-z-z^2+z^4-2*t*z^4): 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
%t A275442 Table[BinCounts[#, {0, 1 + Floor[n/4], 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_, ___} /; a == 2]], 1]]], {n, 0, 20}] // Flatten (* _Michael De Vlieger_, Aug 17 2016 *)
%o A275442 (PARI)
%o A275442 T_zt(max_row) = {my(N = max_row+1, z='z+O('z^N), h=(1-z^2)/(1-z-z^2+z^4-2*t*z^4)); vector(N, n, Vecrev(polcoeff(h, n-1)))}
%o A275442 T_zt(10) \\ _John Tyler Rascoe_, May 09 2025
%Y A275442 Cf. A000931, A005251, A180177, A275433, A275443, A275446.
%K A275442 nonn,easy,tabf
%O A275442 0,4
%A A275442 _Emeric Deutsch_, Aug 16 2016