OFFSET
0,9
COMMENTS
FORMULA
G.f.: G(t,x) = (1-x)/(1 - (1 + t)x - (1 - t)x^3).
EXAMPLE
T(3,2) = 2 because we have [1,2],[2,1].
T(6,3) = 5 because we have [2,2,2],[1,1,1,3],[1,1,3,1],[1,3,1,1],[3,1,1,1].
Triangle begins:
1,
0, 1,
0, 1, 1,
1, 0, 2, 1,
1, 2, 1, 3, 1,
1, 4, 3, 3, 4, 1,
2, 4, 9, 5, 6, 5, 1,
3, 7, 12, 16, 9, 10, 6, 1,
4, 13, 18, 28, 26, 16, 15, 7, 1,
...
MAPLE
g := (1-x)/(1-(1+t)*x-(1-t)*x^3): gser := simplify(series(g, x = 0, 17)): for n from 0 to 15 do p[n] := sort(expand(coeff(gser, x, n))) end do: for n from 0 to 15 do seq(coeff(p[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form
MATHEMATICA
nmax = 12;
s = Series[(1-x)/(1 - (1+t) x - (1-t) x^3), {x, 0, nmax}, {t, 0, nmax}];
T[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 16 2017 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Dec 15 2017
STATUS
approved