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A370649 Dimension of space of equivariant linear maps from R^{n^3} to R^{n^3} under diagonal action of {-1, 1}^n.

%I A370649 #37 Jul 15 2024 16:35:21
%S A370649 0,1,32,183,544,1205,2256,3787,5888,8649,12160,16511,21792,28093,
%T A370649 35504,44115,54016,65297,78048,92359,108320,126021,145552,167003,
%U A370649 190464,216025,243776,273807,306208,341069,378480,418531,461312,506913,555424,606935,661536,719317
%N A370649 Dimension of space of equivariant linear maps from R^{n^3} to R^{n^3} under diagonal action of {-1, 1}^n.
%H A370649 D. Lim, J. Robinson, S. Jegelka, and H. Maron, <a href="https://arxiv.org/abs/2312.02339">Expressive Sign Equivariant Networks for Spectral Geometric Learning</a>, Conference on Neural Information Processing Systems (NeurIPS), 2023. See Table 4 at p. 24.
%H A370649 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A370649 a(n) = (1/2^n) * Sum_{s in {-1,1}^n} (s_1 + s_2 + ... + s_n)^6 [from Proposition 7 of Lim et al.]. - _Sean A. Irvine_, Jul 14 2024
%F A370649 From _Alois P. Heinz_, Jul 14 2024: (Start)
%F A370649 a(n) = 2^(-n) * Sum_{k=0..n} (2*k-n)^6 * binomial(n,k).
%F A370649 G.f.: x*(61*x^2+28*x+1)/(x-1)^4.
%F A370649 a(n) = 15*n^3 - 30*n^2 + 16*n. (End)
%F A370649 E.g.f.: exp(x)*x*(1 + 15*x + 15*x^2). - _Stefano Spezia_, Jul 15 2024
%p A370649 a:= n-> ((15*n-30)*n+16)*n:
%p A370649 seq(a(n), n=0..37);  # _Alois P. Heinz_, Jul 14 2024
%o A370649 (Python)
%o A370649 def A370649(n): return n*(15*(n-1)**2+1) # _Chai Wah Wu_, Jul 15 2024
%Y A370649 Cf. A000567.
%K A370649 nonn,easy
%O A370649 0,3
%A A370649 _Derek Lim_, Feb 25 2024
%E A370649 a(21)-a(33) from _Sean A. Irvine_, Jul 14 2024
%E A370649 a(34)-a(37) from _Alois P. Heinz_, Jul 14 2024