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%I #33 Feb 14 2023 08:58:44
%S 1,6,21,56,125,246,441,736,1161,1750,2541,3576,4901,6566,8625,11136,
%T 14161,17766,22021,27000,32781,39446,47081,55776,65625,76726,89181,
%U 103096,118581,135750,154721,175616,198561,223686,251125,281016,313501,348726,386841
%N Partial sum of centered tetrahedral numbers A005894.
%C From _Robert A. Russell_, Oct 09 2020: (Start)
%C a(n-1) is the number of achiral colorings of the 5 tetrahedral facets (or vertices) of a regular 4-dimensional simplex using n or fewer colors. An achiral arrangement is identical to its reflection. The 4-dimensional simplex is also called a 5-cell or pentachoron. Its Schläfli symbol is {3,3,3}.
%C There are 60 elements in the automorphism group of the 4-dimensional simplex that are not in its rotation group. Each is an odd permutation of the vertices and can be associated with a partition of 5 based on the conjugacy class of the permutation. The first formula for a(n-1) is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C Partition Count Odd Cycle Indices
%C 41 30 x_1x_4^1
%C 32 20 x_2^1x_3^1
%C 2111 10 x_1^3x_2^1 (End)
%H Nathaniel Johnston, <a href="/A132366/b132366.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = (n^4 + 4*n^3 + 11*n^2 + 14*n + 6)/6.
%F G.f.: -(x+1)*(x^2+1) / (x-1)^5. - _Colin Barker_, May 04 2013
%F From _Robert A. Russell_, Oct 09 2020: (Start)
%F a(n-1) = n^2 * (5 + n^2) / 6.
%F a(n-1) = binomial(n+4,5) - binomial(n,5).
%F a(n-1) = 1*C(n,1) + 4*C(n,2) + 6*C(n,3) + 4*C(n,4), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
%F a(n-1) = 2*A000389(n+4) - A337895(n) = A337895(n) - 2*A000389(n) = A000389(n+4) - A000389(n).
%F G.f. for a(n-1): x * (x+1) * (x^2+1) / (1-x)^5. (End)
%F From _Amiram Eldar_, Feb 14 2023: (Start)
%F Sum_{n>=0} 1/a(n) = Pi^2/5 + 3/25 - 3*Pi*coth(sqrt(5)*Pi)/(5*sqrt(5)).
%F Sum_{n>=0} (-1)^n/a(n) = Pi^2/10 - 3/25 + 3*Pi*cosech(sqrt(5)*Pi)/(5*sqrt(5)). (End)
%t Do[Print[n, " ", (n^4 + 4 n^3 + 11 n^2 + 14 n + 6)/6 ], {n, 0, 10000}]
%t Accumulate[Table[(2n+1)(n^2+n+3)/3,{n,0,40}]] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,6,21,56,125},40] (* _Harvey P. Dale_, Feb 26 2020 *)
%Y Cf. A000292, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
%Y Cf. A337895 (oriented), A000389(n+4) (unoriented), A000389 (chiral), A331353 (5-cell edges, faces), A337955 (8-cell vertices, 16-cell facets), A337958 (16-cell vertices, 8-cell facets), A338951 (24-cell), A338967 (120-cell, 600-cell).
%Y a(n-1) = A325001(4,n).
%K easy,nonn
%O 0,2
%A _Jonathan Vos Post_, Nov 09 2007
%E Corrected offset, Mathematica program by Tomas J. Bulka (tbulka(AT)rodincoil.com), Sep 02 2009
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