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

Author

Peter Luschny, May 01 2011

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Peter Luschny, The lost Catalan numbers.

Crossrefs

Cf. A037965, A097070, A056040, A212303.

Programs

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 *)

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

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