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A194588 a(n) = A189912(n-1)-a(n-1) for n>0, a(0) = 1; extended Riordan numbers. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jul 30 2013 *)

CROSSREFS

Cf. A005043, A194589.

Sequence in context: A239682 A009725 A053098 * A175395 A169888 A168506

Adjacent sequences:  A194585 A194586 A194587 * A194589 A194590 A194591

KEYWORD

nonn

AUTHOR

Peter Luschny, Aug 30 2011

STATUS

approved

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Last modified February 23 22:17 EST 2018. Contains 299595 sequences. (Running on oeis4.)