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A194588 a(n) = A189912(n-1)-a(n-1) for n>0, a(0) = 1; extended Riordan numbers.

%I A194588 #15 Jul 30 2013 04:06:02
%S A194588 1,0,2,2,8,17,49,128,356,983,2759,7779,22087,63000,180478,518846,
%T A194588 1496236,4326383,12539335,36419069,105971473,308866226,901573732,
%U A194588 2635235789,7712078755,22594899002,66266698424,194531585078,571561286576,1680679630089,4945738222801
%N A194588 a(n) = A189912(n-1)-a(n-1) for n>0, a(0) = 1; extended Riordan numbers.
%H A194588 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>.
%F A194588 a(n) = ((n+1) mod 2) + (1/2)*sum_{k=1..n}((-1)^k*binomial(n,k)*((k+1)/2)^(k mod 2)*(k+1)$+2*(-1)^n*(2*k)$/(k+1)), where n$ denotes the swinging factorial A056040(n).
%p A194588 A189912 := n -> add(n!/((n-k)!*iquo(k,2)!^2 *(iquo(k,2)+1)),k=0..n):
%p A194588 A194588 := n -> `if`(n=0,1,A189912(n-1)-A194588(n-1)):
%t A194588 a[0] = 1; a[n_] := a[n] = Sum[(n-1)!/((n-k-1)!*Quotient[k, 2]!^2*(1 + Quotient[k, 2])), {k, 0, n-1}] - a[n-1]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jul 30 2013 *)
%Y A194588 Cf. A005043, A194589.
%K A194588 nonn
%O A194588 0,3
%A A194588 _Peter Luschny_, Aug 30 2011