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A307710 a(n) is the determinant of the Vandermonde matrix of the digits of n in factorial base.

%I A307710 #7 Apr 24 2019 00:02:21
%S A307710 1,1,-1,0,-2,-1,0,0,0,0,2,0,0,2,-2,0,0,0,0,6,-6,0,-6,-2,0,0,0,0,0,0,0,
%T A307710 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A307710 0,0,12,-12,0,0,0,0,0,0,0,0,12,0,0,0,0
%N A307710 a(n) is the determinant of the Vandermonde matrix of the digits of n in factorial base.
%C A307710 This sequence is a variant of A307651, and has infinitely many nonzero terms.
%H A307710 Wikipedia, <a href="https://en.wikipedia.org/wiki/Vandermonde_matrix">Vandermonde matrix</a>
%H A307710 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A307710 a(n) != 0 iff n belongs to A321682.
%e A307710                                     | 3^0 3^1 2^2 |
%e A307710 a(22) = a(3*3! + 2*2! + 0*1!) = det | 2^0 2^1 2^2 | = -6.
%e A307710                                     | 0^0 0^1 0^2 |
%o A307710 (PARI) a(n) = my (d=[]); for (r=2, oo, if (n, d=concat(n%r,d); n\=r, return (matdet(matrix(#d, #d, r, c, d[r]^(c-1))))))
%Y A307710 See A307651 for the decimal variant.
%Y A307710 Cf. A108731, A321682.
%K A307710 sign,base
%O A307710 0,5
%A A307710 _Rémy Sigrist_, Apr 23 2019