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A174727 a(n) = A091137(n+1)/(n+1).

%I A174727 #16 Dec 26 2023 12:08:58
%S A174727 2,6,8,180,288,10080,17280,453600,806400,47900160,87091200,
%T A174727 217945728000,402361344000,2241727488000,4184557977600,
%U A174727 2000741783040000,3766102179840000,2838385676206080000,5377993912811520000,1686001091666411520000,3211430650793164800000,423033001181754163200000
%N A174727 a(n) = A091137(n+1)/(n+1).
%C A174727 Previous name: Inverse Akiyama-Tanigawa algorithm. From a column instead of a row. Bernoulli case A164555/A027642. We start from column 1, 1/2, 1/3, 1/4, 1/5 = A000012/A000027. First row: 1) (unreduced) 1, 1/2, 5/12, 9/24, 251/720 = A002657/A091137 (Cauchy from Bernoulli) (*); 2) (reduced) 1, 1/2, 5/12, 3/8, 251/720 = A002208/A002209 (Stirling and Bernoulli). Unreduced second row: 1/2, 1/6, 1/8, 19/180, 27/288, 863/10080 = A141417(n+1)/a(n).
%C A174727 (*) Reference page 56 (first row) and page 36 (upper main diagonal). From J. C. Adams (and Bashforth) numerical integration. See A165313 and A147998. See A002206 logarithm numbers (Gregory).
%D A174727 P. Curtz, Intégration numérique des systèmes différentiels .. . Note 12, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
%F A174727 a(n) = A091137(n+1)/(n+1).
%t A174727 A091137[n_] := A091137[n] = Product[d, {d, Select[ Divisors[n] + 1, PrimeQ]}]*A091137[n-1]; A091137[0] = 1; a[n_] := A091137[n+1]/(n+1); Table[a[n], {n, 0, 18}] (* Jean-François Alcover_, Aug 14 2012 *)
%o A174727 (PARI) f(n) = my(r =1); forprime(p=2, n+1, r*=p^(n\(p-1))); r; \\ A091137
%o A174727 a(n) = f(n+1)/(n+1); \\ _Michel Marcus_, Jun 30 2019
%Y A174727 Cf. A002689, A091137.
%K A174727 nonn
%O A174727 0,1
%A A174727 _Paul Curtz_, Mar 28 2010
%E A174727 Extended up to a(18) by _Jean-François Alcover_, Aug 14 2012
%E A174727 New name and more terms from _Michel Marcus_, Jun 30 2019