OFFSET
1,2
COMMENTS
For n=1, the figure is a line segment with two vertices. For n=2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with four triangular faces. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. Each of its n+1 facets is a regular (n-1)-dimensional simplex. An achiral coloring is the same as its reflection.
LINKS
Robert A. Russell, Table of n, a(n) for n = 1..1275
FORMULA
A(n,k) = binomial(n+k,n+1) - binomial(k,n+1).
A(n,k) = Sum_{j=1..n} A325003(n,j) * binomial(k,j).
A(n,k) = 2*A325000(n,k) - A324999(n,k) = A324999(n,k) - 2*A325000(n,k-n) = A325000(n,k) - A325000(n,k-n).
G.f. for row n: (x - x^(n+1)) / (1-x)^(n+2).
Linear recurrence for row n: A(n,k) = Sum_{j=1..n+1} -binomial(j-n-2,j) * A(n,k-j).
G.f. for column k: (1 - (1-x^2)^k) / (x*(1-x)^k).
EXAMPLE
The array begins with A(1,1):
1 2 3 4 5 6 7 8 9 10 11 12 13 ...
1 4 9 16 25 36 49 64 81 100 121 144 169 ...
1 5 15 34 65 111 175 260 369 505 671 870 1105 ...
1 6 21 56 125 246 441 736 1161 1750 2541 3576 4901 ...
1 7 28 84 210 461 917 1688 2919 4795 7546 11452 16848 ...
1 8 36 120 330 792 1715 3424 6399 11320 19118 31032 48672 ...
1 9 45 165 495 1287 3003 6434 12861 24265 43593 75087 124683 ...
1 10 55 220 715 2002 5005 11440 24309 48610 92323 167740 293215 ...
...
For A(2,2)=4, the triangle may have 0, 1, 2, or 3 edges of one color.
MATHEMATICA
Table[Binomial[d+1, n+1] - Binomial[d+1-n, n+1], {d, 1, 15}, {n, 1, d}] // Flatten
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Robert A. Russell, Mar 23 2019
STATUS
approved