OFFSET
0,3
COMMENTS
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).
A sequence is palindromic if and only if its degree of asymmetry is 0.
Sum(k*T(n,k), k>=0) = A275434(n).
LINKS
Alois P. Heinz, Rows n = 0..250, flattened
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^2)/((1-z)*(1-2*z^2) - 2tz^3). 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
T(4,0) = 4 because we have 4, 22, 121, and 1111.
T(4,1) = 4 because we have 13, 31, 112, and 211.
Triangle starts:
1;
1;
2;
2,2;
4,4;
4,12;
8,20,4.
MAPLE
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
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(n=0, 1, add(b(n-j,
`if`(i=0, j, 0))*`if`(i>0 and i<>j, x, 1), j=1..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..20); # Alois P. Heinz, Jul 29 2016
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabf,changed
AUTHOR
Emeric Deutsch, Jul 29 2016
STATUS
approved