A189911 Row sums of the extended Catalan triangle A189231.

1, 2, 4, 9, 18, 40, 80, 175, 350, 756, 1512, 3234, 6468, 13728, 27456, 57915, 115830, 243100, 486200, 1016158, 2032316, 4232592, 8465184, 17577014, 35154028, 72804200, 145608400, 300874500, 601749000, 1240940160, 2481880320, 5109183315, 10218366630

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Peter Luschny, The lost Catalan numbers.

Formula

Let a = Gamma(n-floor(n/2)), b = Gamma(floor(n/2+3/2)), d = Gamma( floor(n/2+1))^2, c = Gamma(n+1). Then a(n) = c*(a*b+d)/(a*b*d).

a(n) = A162246(n,n) + A162246(n,n+1) for n > 0.

From Peter Luschny, Oct 24 2013 : (Start)

E.g.f.: (x+1)*(BesselI(0, 2*x)+BesselI(1, 2*x)).

O.g.f.: I*(2*x^2-1)/(2*sqrt(2*x+1)*x*(2*x-1)^(3/2))-1/(2*x).

Recurrence: a(0) = 1; a(n) = a(n-1)*2 if n is even else ([n/2]+2)*(2*[n/2]+1)/([n/2]+1)^2. ([.] the floor brackets.)

a(n) = A056040(n) + A212303(n) = n$*(1+[(n+1)/2]^((-1)^n)), where n$ is the swinging factorial.

a(2*n) = (n+1)*C(2*n, n) (A037965);

a(2*n+1) = (n+2)*C(2*n+1, n+1) (A097070). (End)

Sum_{n>=0} 1/a(n) = 4*Pi/sqrt(3) - Pi^2/3 - 2. - Amiram Eldar, Aug 20 2022

D-finite with recurrence: (n-2)*(n+1)^2*a(n) - (2*(n-2)^2+2*n-12)*a(n-1) - 4*(n+2)*(n-1)^2*a(n-2) = 0. - Georg Fischer, Nov 25 2022

Maple

A189911 := proc(n) local a, b, d; if n = 0 then 1 else
a := GAMMA(n-floor(n/2)); b := GAMMA(floor(n/2+3/2));
d := GAMMA(floor(n/2+1))^2; GAMMA(n+1)*(a*b+d)/(a*b*d) fi end: seq(A189911(n), n=0..32);
A189911 := proc(n) h:=irem(n, 2); g:=iquo(n, 2); (g+h+1)*binomial(2*g+h, g+h) end; # Peter Luschny, Oct 24 2013

Mathematica

a[n_] := Module[{q, r}, {q, r} = QuotientRemainder[n, 2]; (q+r+1)*Pochhammer[q+1, q+r]/(q+r)!]; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jan 09 2014 *)

Prog

(Sage)
def A189911():
    r, n = 1, 1
    while True:
        yield r
        h = n//2
        r *= 2 if is_even(n) else (h+2)*(2*h+1)/(h+1)^2
        n += 1
a = A189911(); [next(a) for i in range(16)]  # Peter Luschny, Oct 24 2013

Crossrefs

Cf. A037965, A097070, A056040, A212303.

Sequence in context: A289846 A193201 A038044 * A026732 A171003 A094291

Adjacent sequences: A189908 A189909 A189910 * A189912 A189913 A189914

Keyword

nonn

Author

Peter Luschny, May 01 2011

Status

approved