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A273194 a(n) = numerator(R(n,3)), where R(n,d) = (Product_{j prime to d} Pochhammer(j/d, n)) / n!.

%I A273194 #17 Dec 12 2023 18:39:19
%S A273194 1,2,20,1120,30800,1121120,152472320,8277068800,523524601600,
%T A273194 340290991040000,27631628472448000,2491870494969856000,
%U A273194 741331472253532160000,80177849999112785920000,9392262428467497779200000,3554032102932101159649280000,480238587908700169197608960000
%N A273194 a(n) = numerator(R(n,3)), where R(n,d) = (Product_{j prime to d} Pochhammer(j/d, n)) / n!.
%C A273194 Also the numerators of the nonzero coefficients in the expansion of hypergeom([Seq_{k=1..m-1} k/m], [], (x/m)^m) for m = 3.
%p A273194 Hlist := proc(m,size) local H, S;
%p A273194 H := m -> hypergeom([seq(k/m, k=1..m-1)], [], (x/m)^m);
%p A273194 S := m -> series(H(m), x, (m+1)*size);
%p A273194 seq(numer(coeff(S(m), x, m*n)), n=0..size) end:
%p A273194 A273194_list := size -> Hlist(3, size);
%p A273194 # Alternative:
%p A273194 coprimes := n -> select(j -> igcd(j, n) = 1, {$1..n}):
%p A273194 R := (n, d) -> mul(pochhammer(j/d, n), j in coprimes(d)) / n!:
%p A273194 seq(numer(R(n, 3)), n = 0..16); # _Peter Luschny_, May 20 2021
%Y A273194 R(n, 1) = A000012 / A000012.
%Y A273194 R(n, 2) = A001790 / A046161.
%Y A273194 R(n, 3) = (this sequence) / A344402.
%Y A273194 Cf. A273192, A273193.
%K A273194 nonn,frac
%O A273194 0,2
%A A273194 _Peter Luschny_, Jun 06 2016