OFFSET
0,2
COMMENTS
The complete truncation of a 1-dimensional segment is also a 1-dimensional segement (rather than degenerating to a point).
FORMULA
G.f. for rows (n > 0): (((x+1)^n-1)*(x+n+2))/x-n-binomial(n+1,2)*(x+1).
O.g.f.: (1/(1-(x+1)*y)^2-(x+1)/(1-y)^2)/x + 1/((1-(x+1)*y)*(1-y))+1+y*(x+1)*(1-1/(1-y)^3).
E.g.f.: ((x+1)*(z+1)+1)*exp(z)*(exp(x*z)-1)/x + 1 - (x+1)*z*((z+2)*exp(z)-2)/2.
EXAMPLE
Triangle begins:
1;
2, 1;
3, 3, 1;
6, 12, 8, 1;
10, 30, 30, 10, 1;
...
Row 2 describes a triangle.
Row 3 describes an octahedron.
MATHEMATICA
Flatten[Table[
CoefficientList[
D[((x + 1) (z + 1) + 1) Exp[z] (Exp[x z] - 1)/x +
1 - (x + 1) z ((z + 2)*Exp[z] - 2)/2, {z, k}] /. z -> 0, x], {k, 0,
10}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vincent J. Matsko, Apr 02 2016
STATUS
approved