%I #13 Jun 26 2019 07:54:04

%S 0,0,1,-2,7,-14,38,-76,187,-374,874,-1748,3958,-7916,17548,-35096,

%T 76627,-153254,330818,-661636,1415650,-2831300,6015316,-12030632,

%U 25413342,-50826684,106853668,-213707336,447472972,-894945944,1867450648,-3734901296,7770342787

%N a(n) = (-1)^n*(A056040(n+1)*A152271(n)-2^n)/2.

%C The binomial transform of a(n) are the complementary Riordan numbers A194589 (see link).

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*cr(k), where cr(k) are the complementary Riordan numbers A194589.

%p A056040 := n -> n!/iquo(n,2)!^2:

%p A152271 := n -> `if`(n mod 2 = 0, 1, (n+1)/2):

%p A194590 := n -> (-1)^n*(A056040(n+1)*A152271(n)-2^n)/2:

%t sf[n_] := n!/Quotient[n, 2]!^2;

%t a[n_] := (-1)^n (sf[n + 1] * If[EvenQ[n], 1, (n + 1)/2] - 2^n)/2;

%t Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Jun 26 2019 *)

%Y Cf. A107373 (has offset 1).

%K sign

%O 0,4

%A _Peter Luschny_, Aug 30 2011