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%I #26 Oct 21 2020 03:48:50
%S 1,3,1,6,4,1,10,10,5,1,15,20,15,6,1,21,35,35,21,7,1,28,56,70,56,28,8,
%T 1,36,84,126,126,84,36,9,1,45,120,210,252,210,120,45,10,1,55,165,330,
%U 462,462,330,165,55,11,1,66,220,495,792,924,792,495,220,66,12,1
%N Array read by descending antidiagonals: T(n,k) is the number of unoriented colorings of the facets (or vertices) of a regular n-dimensional simplex using up to k colors.
%C 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 unoriented colorings are the same if congruent; chiral pairs are counted as one.
%C Note that antidiagonals are part of rows of the Pascal triangle.
%C T(n,k-n) is the number of chiral pairs of colorings of the facets (or vertices) of a regular n-dimensional simplex using k or fewer colors. - _Robert A. Russell_, Sep 28 2020
%H Robert A. Russell, <a href="/A325000/b325000.txt">Table of n, a(n) for n = 1..1275</a>
%F T(n,k) = binomial(n+k,n+1) = A007318(n+k,n+1).
%F T(n,k) = Sum_{j=1..n+1} A007318(n,j-1) * binomial(k,j).
%F T(n,k) = A324999(n,k) + T(n,k-n) = (A324999(n,k) - A325001(n,k)) / 2 = T(n,k-n) + A325001(n,k). - _Robert A. Russell_, Sep 28 2020
%F G.f. for row n: x / (1-x)^(n+2).
%F Linear recurrence for row n: T(n,k) = Sum_{j=1..n+2} -binomial(j-n-3,j) * T(n,k-j).
%F G.f. for column k: (1 - (1-x)^k) / (x * (1-x)^k) - k.
%F T(n,k-n) = A324999(n,k) - T(n,k) = (A324999(n,k) - A325001(n,k)) / 2 = T(n,k) - A325001(n,k). - _Robert A. Russell_, Oct 10 2020
%e The array begins with T(1,1):
%e 1 3 6 10 15 21 28 36 45 55 66 78 91 105 ...
%e 1 4 10 20 35 56 84 120 165 220 286 364 455 560 ...
%e 1 5 15 35 70 126 210 330 495 715 1001 1365 1820 2380 ...
%e 1 6 21 56 126 252 462 792 1287 2002 3003 4368 6188 8568 ...
%e 1 7 28 84 210 462 924 1716 3003 5005 8008 12376 18564 27132 ...
%e 1 8 36 120 330 792 1716 3432 6435 11440 19448 31824 50388 77520 ...
%e 1 9 45 165 495 1287 3003 6435 12870 24310 43758 75582 125970 203490 ...
%e 1 10 55 220 715 2002 5005 11440 24310 48620 92378 167960 293930 497420 ...
%e ...
%e For T(1,2) = 3, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses two colors. For T(2,2)=4, the triangle may have 0, 1, 2, or 3 edges of one color.
%t Table[Binomial[d+1,n+1], {d,1,15}, {n,1,d}] // Flatten
%Y Cf. A324999 (oriented), A325001 (achiral).
%Y Unoriented: A007318(n,k-1) (exactly k colors), A327084 (edges, ridges), A337884 (faces, peaks), A325005 (orthotope facets, orthoplex vertices), A325013 (orthoplex facets, orthotope vertices).
%Y Chiral: A327085 (edges, ridges), A337885 (faces, peaks), A325006 (orthotope facets, orthoplex vertices), A325014 (orthoplex facets, orthotope vertices).
%Y Cf. A104712 (same sequence for a triangle; same sequence apart from offset).
%Y Rows 1-4 are A000217, A000292, A000332(n+3), A000389(n+4). - _Robert A. Russell_, Sep 28 2020
%K nonn,tabl,easy
%O 1,2
%A _Robert A. Russell_, Mar 23 2019
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