A203228 -id:A203228 - cp's OEIS frontend

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

Clark Kimberling, Dec 30 2011

nonn

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

			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

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

Cf. A203228, A000142, A000178.

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

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 *)

A204245 Determinant of the n-th principal submatrix of A204244.

Original entry on oeis.org

1, 1, 4, 84, 9792, 7015680, 35334144000, 1424547274752000, 516934658477260800000, 1875850653748811739955200000, 74877948716953984356707205120000000, 35866656974348638336845775173058560000000000

Offset: 1

Clark Kimberling, Jan 13 2012

nonn

Cf. A204244, A203228.

f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, i_] := i!;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 12}, {i, 1, n}]]      (* A204244 *)
Table[Det[m[n]], {n, 1, 15}]   (* A204245 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 14}]   (*  A203228 *)

A204244 Symmetric matrix given by f(i,1)=1, f(1,j)=1, f(i,i)=i! and f(i,j)=0 otherwise.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 0, 0, 1, 1, 0, 6, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 24, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 120, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 720, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 5040, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0

Offset: 1

Clark Kimberling, Jan 13 2012

			Northwest corner:
1 1 1 1
1 2 0 0
1 0 6 0
1 0 0 24

Cf. A204245, A203228.

f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, i_] := i!;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 12}, {i, 1, n}]]      (* A204244 *)
Table[Det[m[n]], {n, 1, 15}]   (* A204245 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 14}]   (*  A203228 *)

cp's OEIS Frontend

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

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A204245 Determinant of the n-th principal submatrix of A204244.

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A204244 Symmetric matrix given by f(i,1)=1, f(1,j)=1, f(i,i)=i! and f(i,j)=0 otherwise.

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