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

1, 2, 5, 32, 780, 93888, 67633920, 340899840000, 13745206960128000, 4987865758275993600000, 18099969098565397826764800000, 722492853172221856076141690880000000, 346075232923849611911833538569175040000000000

Offset: 1

Author

Clark Kimberling, Dec 30 2011

Keywords

• nonn

Comments

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

a(n)/A000178(n-1) -> e as n -> oo. - Daniel Suteu, Jul 30 2016

Examples

			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
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..44

Crossrefs

Cf. A203228, A000142, A000178.

Programs

Maple

a:= n-> coeff(mul(i!*x+1, i=0..n-1), x, n-1):
seq(a(n), n=1..15);  # Alois P. Heinz, Sep 08 2019

Mathematica

f[k_] := (k - 1)!; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 14}]
Flatten[{1, Table[Det[Table[BellB[i+j], {i, n}, {j, n}]], {n, 1, 15}]}] (* Vaclav Kotesovec, Nov 28 2016 *)