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

1, 0, 2, 2, 8, 17, 49, 128, 356, 983, 2759, 7779, 22087, 63000, 180478, 518846, 1496236, 4326383, 12539335, 36419069, 105971473, 308866226, 901573732, 2635235789, 7712078755, 22594899002, 66266698424, 194531585078, 571561286576, 1680679630089, 4945738222801

Offset: 0

Author

Peter Luschny, Aug 30 2011

Keywords

• nonn

Links

• Peter Luschny, The lost Catalan numbers.

Crossrefs

Cf. A005043, A194589.

Programs

Maple

A189912 := n -> add(n!/((n-k)!*iquo(k,2)!^2 *(iquo(k,2)+1)),k=0..n):
A194588 := n -> `if`(n=0,1,A189912(n-1)-A194588(n-1)):

Mathematica

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 *)

Formula

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).