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A275433 Triangle read by rows: T(n,k) is the number of compositions of n having degree of asymmetry equal to k (n>=0; 0<=k<=n/3).

%I A275433 #30 Apr 11 2025 07:58:02
%S A275433 1,1,2,2,2,4,4,4,12,8,20,4,8,44,12,16,68,44,16,132,100,8,32,196,252,
%T A275433 32,32,356,500,136,64,516,1068,384,16,64,900,1956,1096,80,128,1284,
%U A275433 3804,2592,384,128,2180,6612,6152,1280,32,256,3076,12108,13056,4080,192,256,5124,20292,27784,11056,1024,512,7172,35644,54816,28960,3904,64
%N A275433 Triangle read by rows: T(n,k) is the number of compositions of n having degree of asymmetry equal to k (n>=0; 0<=k<=n/3).
%C A275433 The degree of asymmetry of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the degree of asymmetry of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).
%C A275433 A sequence is palindromic if and only if its degree of asymmetry is 0.
%C A275433 Sum_{k>=0} k*T(n,k) = A275434(n).
%H A275433 Alois P. Heinz, <a href="/A275433/b275433.txt">Rows n = 0..250, flattened</a>
%H A275433 Sergi Elizalde and Emeric Deutsch, <a href="https://doi.org/10.54550/ECA2022V2S1R7">The degree of asymmetry of a sequence</a>, Enum. Combinat. Applic. 2 (2002) no 1 #S2R7 eq. (1).
%H A275433 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 A275433 G.f.: G(t,z) = (1-z^2)/((1-z)*(1-2*z^2) - 2*t*z^3). 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 A275433 T(4,0) = 4 because we have 4, 22, 121, and 1111.
%e A275433 T(4,1) = 4 because we have 13, 31, 112, and 211.
%e A275433 Triangle starts:
%e A275433   1;
%e A275433   1;
%e A275433   2;
%e A275433   2,2;
%e A275433   4,4;
%e A275433   4,12;
%e A275433   8,20,4.
%p A275433 G := (1-z^2)/((1-z)*(1-2*z^2)-2*t*z^3): Gser := simplify(series(G, z = 0, 24)): 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
%p A275433 # second Maple program:
%p A275433 b:= proc(n, i) option remember; expand(`if`(n=0, 1, add(b(n-j,
%p A275433       `if`(i=0, j, 0))*`if`(i>0 and i<>j, x, 1), j=1..n)))
%p A275433     end:
%p A275433 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
%p A275433 seq(T(n), n=0..20);  # _Alois P. Heinz_, Jul 29 2016
%t A275433 b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, Sum[b[n - j, If[i == 0, j, 0]]*If[i > 0 && i != j, x, 1], {j, 1, n}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Dec 22 2016, after _Alois P. Heinz_ *)
%Y A275433 Cf. A275434.
%Y A275433 Row sums give A011782.
%Y A275433 Column k=0 gives A016116.
%K A275433 nonn,tabf
%O A275433 0,3
%A A275433 _Emeric Deutsch_, Jul 29 2016