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

%I A320962 #16 Nov 10 2018 16:36:16
%S A320962 0,1,-2,10,-90,1248,-24360,631440,-20865600,852647040,-42085008000,
%T A320962 2462394816000,-168193308729600,13241729554099200,
%U A320962 -1188734048799897600,120563962753538304000,-13704613258628388096000,1733764260005567741952000,-242606144946628642443264000
%N A320962 a(n) = (-1)^(n-1)*(n-1)!*Sum_{i=0..n} Stirling2(n, i) if n > 0 and 0 otherwise.
%C A320962 Also the exponential limit as defined in A320956 of log(x + 1).
%e A320962 Illustration of the convergence in the sense of A320956:
%e A320962   [0] 0, 0,  0,  0,   0,    0,      0,      0,         0, ...
%e A320962   [1] 0, 1, -1,  2,  -6,   24,   -120,    720,     -5040, ... A133942
%e A320962   [2] 0, 1, -2,  8, -48,  384,  -3840,  46080,   -645120, ... A000165
%e A320962   [3] 0, 1, -2, 10, -84,  984, -14640, 262800,  -5513760, ... A321398
%e A320962   [4] 0, 1, -2, 10, -90, 1224, -22440, 514800, -14086800, ...
%e A320962   [5] 0, 1, -2, 10, -90, 1248, -24240, 615600, -19378800, ...
%e A320962   [6] 0, 1, -2, 10, -90, 1248, -24360, 630720, -20719440, ...
%e A320962   [7] 0, 1, -2, 10, -90, 1248, -24360, 631440, -20860560, ...
%e A320962   [8] 0, 1, -2, 10, -90, 1248, -24360, 631440, -20865600, ...
%p A320962 a := n -> `if`(n=0, 0, (-1)^(n-1)*(n-1)!*add(Stirling2(n, i), i=0..n)):
%p A320962 seq(a(n), n=0..19);
%p A320962 # Alternatively use the function ExpLim defined in A320956.
%p A320962 ExpLim(19, x -> ln(x+1));
%t A320962 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 *)
%o A320962 (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
%Y A320962 Cf. A320956, A133942 (n=1), A000165 (n=2), A321398 (n=3).
%K A320962 sign
%O A320962 0,3
%A A320962 _Peter Luschny_, Nov 07 2018