OFFSET
1,8
COMMENTS
LINKS
Alois P. Heinz, Rows n = 1..350, flattened
FORMULA
G.f.: G(t,x) = -1+product(1+tx^j+t^2*x^(2j)+t^3*x^(3j), j=1..infinity).
EXAMPLE
T(7,3) = 4 because we have [5,1,1], [4,2,1], [3,3,1] and [3,2,2].
Triangle starts:
1;
1, 1;
1, 1, 1;
1, 2, 1;
1, 2, 2, 1;
1, 3, 3, 2;
1, 3, 4, 3, 1;
MAPLE
g:=-1+product(1+t*x^j+t^2*x^(2*j)+t^3*x^(3*j), j=1..35): gser:=simplify(series(g, x=0, 23)): for n from 1 to 18 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 18 do seq(coeff(P[n], t^j), j=1..floor(sqrt(6*n+6)-3/2)) od; # yields sequence in triangular form
# second Maple program
b:= proc(n, i) option remember; local j; if n=0 then 1
elif i<1 then 0 else []; for j from 0 to min(3, n/i) do
zip((x, y)->x+y, %, [0$j, b(n-i*j, i-1)], 0) od; %[] fi
end:
T:= n-> subsop(1=NULL, [b(n, n)])[]:
seq(T(n), n=1..20); # Alois P. Heinz, Jan 08 2013
MATHEMATICA
max = 18; g = -1+Product[1+t*x^j+t^2*x^(2j)+t^3*x^(3j), {j, 1, max}]; t[n_, k_] := SeriesCoefficient[g, {x, 0, n}, {t, 0, k}]; Table[DeleteCases[Table[t[n, k], {k, 1, n}], 0], {n, 1, max}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Mar 07 2006
STATUS
approved