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A203227 (n-1)-st elementary symmetric function of (0!,...,(n-1)!).

%I A203227 #24 Jun 02 2025 07:33:51
%S A203227 1,2,5,32,780,93888,67633920,340899840000,13745206960128000,
%T A203227 4987865758275993600000,18099969098565397826764800000,
%U A203227 722492853172221856076141690880000000,346075232923849611911833538569175040000000000
%N A203227 (n-1)-st elementary symmetric function of (0!,...,(n-1)!).
%C A203227 Each term appears as an unreduced numerator in the following partial infinite sum: f(0) = 1; f(n) = f(n-1)/n; Sum_{k>=0}(f(k)) = e. - _Daniel Suteu_, Jul 30 2016
%C A203227 a(n)/A000178(n-1) -> e as n -> oo. - _Daniel Suteu_, Jul 30 2016
%H A203227 Alois P. Heinz, <a href="/A203227/b203227.txt">Table of n, a(n) for n = 1..44</a>
%e A203227 For n=4, the 3rd elementary symmetric polynomial in the 4 variables a, b, c, and d is abc + abd + acd + bcd. Setting a = 0! = 1, b = 1! = 1, c = 2! = 2, and d = 3! = 6 gives a(4) = 1*1*2 + 1*1*6 + 1*2*6 + 1*2*6 = 2 + 6 + 12 + 12 = 32. - _Michael B. Porter_, Aug 17 2016
%p A203227 a:= n-> coeff(mul(i!*x+1, i=0..n-1), x, n-1):
%p A203227 seq(a(n), n=1..15);  # _Alois P. Heinz_, Sep 08 2019
%t A203227 f[k_] := (k - 1)!; t[n_] := Table[f[k], {k, 1, n}]
%t A203227 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A203227 Table[a[n], {n, 1, 14}]
%t A203227 Flatten[{1, Table[Det[Table[BellB[i+j], {i, n}, {j, n}]], {n, 1, 15}]}] (* _Vaclav Kotesovec_, Nov 28 2016 *)
%Y A203227 Cf. A203228, A000142, A000178.
%K A203227 nonn
%O A203227 1,2
%A A203227 _Clark Kimberling_, Dec 30 2011