A320962 a(n) = (-1)^(n-1)*(n-1)!*Sum_{i=0..n} Stirling2(n, i) if n > 0 and 0 otherwise.

0, 1, -2, 10, -90, 1248, -24360, 631440, -20865600, 852647040, -42085008000, 2462394816000, -168193308729600, 13241729554099200, -1188734048799897600, 120563962753538304000, -13704613258628388096000, 1733764260005567741952000, -242606144946628642443264000

Offset: 0

Author

Peter Luschny, Nov 07 2018

Keywords

• sign

Comments

Also the exponential limit as defined in A320956 of log(x + 1).

Examples

			Illustration of the convergence in the sense of A320956:
  [0] 0, 0,  0,  0,   0,    0,      0,      0,         0, ...
  [1] 0, 1, -1,  2,  -6,   24,   -120,    720,     -5040, ... A133942
  [2] 0, 1, -2,  8, -48,  384,  -3840,  46080,   -645120, ... A000165
  [3] 0, 1, -2, 10, -84,  984, -14640, 262800,  -5513760, ... A321398
  [4] 0, 1, -2, 10, -90, 1224, -22440, 514800, -14086800, ...
  [5] 0, 1, -2, 10, -90, 1248, -24240, 615600, -19378800, ...
  [6] 0, 1, -2, 10, -90, 1248, -24360, 630720, -20719440, ...
  [7] 0, 1, -2, 10, -90, 1248, -24360, 631440, -20860560, ...
  [8] 0, 1, -2, 10, -90, 1248, -24360, 631440, -20865600, ...
		

Crossrefs

Cf. A320956, A133942 (n=1), A000165 (n=2), A321398 (n=3).

Programs

Maple

a := n -> `if`(n=0, 0, (-1)^(n-1)*(n-1)!*add(Stirling2(n, i), i=0..n)):
seq(a(n), n=0..19);
# Alternatively use the function ExpLim defined in A320956.
ExpLim(19, x -> ln(x+1));

Mathematica

a[n_] := If[n == 0, 0, (-1)^(n - 1)*(n - 1)!*Sum[StirlingS2[n, i], {i, 0, n}]]; Array[a, 19, 0] (* Amiram Eldar, Nov 07 2018 *)

PARI

a(n) = if (n>0, (-1)^(n-1)*(n-1)!*sum(i=0, n, stirling(n, i, 2)), 0); \\ Michel Marcus, Nov 07 2018