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

0,3

Links

Table of n, a(n) for n=0..30.

Peter Luschny, The lost Catalan numbers.

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).

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 *)

Crossrefs

Cf. A005043, A194589.

Sequence in context: A009725 A053098 A354117 * A175395 A169888 A168506

Adjacent sequences: A194585 A194586 A194587 * A194589 A194590 A194591

Keyword

nonn

Author

Peter Luschny, Aug 30 2011

Status

approved