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%I #19 Jun 27 2016 17:23:25
%S 1,0,2,-12,122,-1620,26882,-536172,12506762,-334261380,10075002962,
%T -338180323932,12512502202202,-505992958647540,22204726014875042,
%U -1050993549782729292,53373431773793542442,-2894886293042487680100,167021024758368026331122
%N Apply the "Stirling-Bernoulli transform" to Catalan numbers.
%C The "Stirling-Bernoulli transform" maps a sequence b_0, b_1, b_2, ... to a sequence c_0, c_1, c_2, ..., where if B has o.g.f. B(x), c has e.g.f. exp(x)*B(1-exp(x)). More explicitly, c_n = Sum_{m=0..n} (-1)^m*m!*Stirling2(n+1,m+1)*b_m.
%H Alois P. Heinz, <a href="/A051782/b051782.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k = 0..n} A163626(n,k)*A000108(k). - _Philippe Deléham_, May 25 2015
%p a:= n-> add((-1)^k *k! *Stirling2(n+1, k+1)*binomial(2*k, k)/
%p (k+1), k=0..n):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, May 17 2013
%t a[n_] := Sum[(-1)^k k! StirlingS2[n+1, k+1] CatalanNumber[k], {k, 0, n}];
%t Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Apr 06 2016 *)
%Y Cf. A000108, A163626.
%K sign
%O 0,3
%A _N. J. A. Sloane_, Dec 09 1999
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