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. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.
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+1} A325002(n,j) * binomial(k,j).
A(n,k) = A325000(n,k) + A325000(n,k-n) = 2*A325000(n,k) - A325001(n,k) = 2*A325000(n,k-n) + A325001(n,k).
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+2} -binomial(j-n-3,j) * A(n,k-j).
G.f. for column k: (1 - 2*(1-x)^k + (1-x^2)^k) / (x*(1-x)^k) - 2*k.
EXAMPLE
The array begins with A(1,1):
1 4 9 16 25 36 49 64 81 100 121 144 169 196 ...
1 4 11 24 45 76 119 176 249 340 451 584 741 924 ...
1 5 15 36 75 141 245 400 621 925 1331 1860 2535 3381 ...
1 6 21 56 127 258 483 848 1413 2254 3465 5160 7475 10570 ...
1 7 28 84 210 463 931 1744 3087 5215 8470 13300 20280 30135 ...
1 8 36 120 330 792 1717 3440 6471 11560 19778 32616 52104 80952 ...
1 9 45 165 495 1287 3003 6436 12879 24355 43923 76077 127257 206493 ...
1 10 55 220 715 2002 5005 11440 24311 48630 92433 168180 294645 499422 ...
...
For A(1,2) = 4, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses two colors. 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