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%I #35 Mar 10 2023 07:25:20
%S 1,2802750,105904482864,187226450755016,61150982606571900,
%T 6737855626357107000,342689297671355738880,9659365383584921484480,
%U 169366933728740293383600,1995772772375467764487200
%N Inequivalent simultaneous colorings of the faces, vertices and edges of the cube under rotational symmetry using exactly n colors.
%C This sequence is finite as we have slots for at most 26 colors.
%H Marko Riedel et al., Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/4516333/">Coloring faces, vertices, edges of a cube</a>.
%F a(n) = Sum_{q=0..n} C(n,q) (-1)^(n-q) A356685(q).
%F a(n) = (n!/24) * (S2(26,n) + 9*S2(14,n) + 8*S2(10,n) + 6*S2(8,n)) where S2 is the Stirling number of the second kind (Stirling set number).
%F Cycle index is (1/24) * (x1^26 + 6*x1^2*x4^6 + 9*x1^2*x2^12 + 8*x1^2*x3^8).
%Y Cf. A356685.
%K nonn,fini
%O 1,2
%A _Marko Riedel_, Aug 22 2022
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