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%I #30 Oct 22 2015 05:15:50
%S 0,1,1,-2,-1,11,-18,-41,317,-680,-1767,19911,-68264,-59643,2076973,
%T -11905466,18577387,269836343,-2819431570,12357816867,17355428041,
%U -752675321800,6318046208653,-21416130683133,-152569023028272,3016508107668601,-23667435182395287
%N a(n) = Sum_{k=0..n} (-1)^(n-k)*k*Stirling2(n,k).
%H Robert Israel, <a href="/A101851/b101851.txt">Table of n, a(n) for n = 0..580</a>
%F E.g.f.: (1-exp(-x))*exp(1-exp(-x)): G.f.: Sum(k*x^k/Product(1+l*x, l = 1 .. k), k = 1 .. infinity).
%F a(n) = Sum_{k=0..n} (-1)^(k+1)*binomial(n,k)*A000587(k+2). - _Peter Luschny_, Apr 17 2011
%F G.f.: x*G(0)/(1+x) where G(k) = 1 + 2*x*(k+1)/((2*k+1)*(2*x*k+2*x+1) - x*(2*k+1)*(2*k+3)*(2*x*k+2*x+1)/(x*(2*k+3) + 2*(k+1)*(2*x*k+3*x+1)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 19 2012
%F G.f.: 1/x - G(0)/x, where G(k) = 1 - x^2*(k+1)/(x^2*(k+1) + (x*k + 1 - x)*(x*k + 1)/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Feb 06 2014
%F a(n) = (-1)^n*(A000587(n)+A000587(n+1)). - _Vladimir Reshetnikov_, Oct 21 2015
%p A101851 := proc(n) local k;
%p add((-1)^(n-k)*k*combinat[stirling2](n,k), k = 0..n) end:
%p seq(A101851(n),n = 0..26); # _Peter Luschny_, Apr 17 2011
%t Table[Sum[(-1)^(n-k) k StirlingS2[n,k],{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Aug 09 2013 *)
%t Table[(-1)^n (BellB[n, -1] + BellB[n + 1, -1]), {n, 0, 25}] (* _Vladimir Reshetnikov_, Oct 21 2015 *)
%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*k*stirling(n, k, 2)); \\ _Michel Marcus_, Oct 22 2015
%Y Cf. A005493, A000587.
%K easy,sign
%O 0,4
%A _Vladeta Jovovic_, Jan 27 2005
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