OFFSET
0,3
COMMENTS
Conjectured asymptotic: a(n) ~ (n^2 + 6*n - 12)/12. - Vladimir Reshetnikov, Dec 28 2021
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..800
EXAMPLE
The slice for n=4 is
1
4 4
6 12 6
4 12 12 4
1 4 6 4 1
with distinct entries 1,4,6,12, so a(4) = 4.
MAPLE
p:= proc(i, j, k) option remember;
if i<0 or j<0 or k<0 or i>k or j>i then 0
elif {i, j, k}={0} then 1
else p(i, j, k-1) +p(i-1, j, k-1) +p(i-1, j-1, k-1)
fi
end:
a:= n-> nops({seq(seq(p(i, j, n), j=0..i), i=0..n)}):
seq(a(n), n=0..50); # Alois P. Heinz, Aug 14 2012
#
seq(nops({coeffs(expand((x+y+z)^n))}), n = 0 .. 100); # César Eliud Lozada, Jul 02 2015
#
seq(nops({seq(seq(n!/(i!*j!*(n-i-j)!), j=i..(n-i)/2), i=0..n/3)}), n=0..100); # Robert Israel, Jul 02 2015
MATHEMATICA
Table[Length @ Union @ Flatten @ Table[Table[n!/(i!*j!*(n-i-j)!), {j, i, (n-i)/2}], {i, 0, n/3}], {n, 0, 100}] (* Jean-François Alcover, Mar 19 2019, after Robert Israel *)
PROG
(PARI) { pt=vector(40, i, matrix(i, i)); pt[1][1, 1]=1; pt[2][1, 1]=1; pt[2][2, 1]=1; pt[2][2, 2]=1; pt[3][1, 1]=1; pt[3][2, 1]=2; pt[3][2, 2]=2; pt[3][3, 1]=1; pt[3][3, 2]=2; pt[3][3, 3]=1; for (i=4, 40, for (j=2, i-1, pt[i][j, 1]=pt[i-1][j-1, 1]+pt[i-1][j, 1]; pt[i][j, j]=pt[i][j, 1]; pt[i][i, j]=pt[i][j, 1] ); pt[i][1, 1]=1; pt[i][i, 1]=1; pt[i][i, i]=1; for(j=3, i-1, for (k=2, j-1, pt[i][j, k]=pt[i-1][j, k]+pt[i-1][j-1, k]+pt[i-1][j-1, k-1]))); pt } { makept(x)=local(xl, v, vc, uc); xl=length(x); v=vector(xl*(xl+1)/2); vc=0; for (i=1, xl, for (j=1, i, v[vc++ ]=x[i, j])); v=vecsort(v); uc=1; for (i=2, length(v), if (v[i]!=v[i-1], uc++)); print1(", "uc) } for (i=1, 40, makept(pt([i]))
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon Perry, Jul 31 2003
EXTENSIONS
More terms from Alois P. Heinz, Aug 14 2012
STATUS
approved