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A189911 Row sums of the extended Catalan triangle A189231.

%I A189911 #29 Nov 25 2022 03:58:43
%S A189911 1,2,4,9,18,40,80,175,350,756,1512,3234,6468,13728,27456,57915,115830,
%T A189911 243100,486200,1016158,2032316,4232592,8465184,17577014,35154028,
%U A189911 72804200,145608400,300874500,601749000,1240940160,2481880320,5109183315,10218366630
%N A189911 Row sums of the extended Catalan triangle A189231.
%H A189911 G. C. Greubel, <a href="/A189911/b189911.txt">Table of n, a(n) for n = 0..1000</a>
%H A189911 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>.
%F A189911 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).
%F A189911 a(n) = A162246(n,n) + A162246(n,n+1) for n > 0.
%F A189911 From _Peter Luschny_, Oct 24 2013 : (Start)
%F A189911 E.g.f.: (x+1)*(BesselI(0, 2*x)+BesselI(1, 2*x)).
%F A189911 O.g.f.: I*(2*x^2-1)/(2*sqrt(2*x+1)*x*(2*x-1)^(3/2))-1/(2*x).
%F A189911 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.)
%F A189911 a(n) = A056040(n) + A212303(n) = n$*(1+[(n+1)/2]^((-1)^n)), where n$ is the swinging factorial.
%F A189911 a(2*n) = (n+1)*C(2*n, n) (A037965);
%F A189911 a(2*n+1) = (n+2)*C(2*n+1, n+1) (A097070). (End)
%F A189911 Sum_{n>=0} 1/a(n) = 4*Pi/sqrt(3) - Pi^2/3 - 2. - _Amiram Eldar_, Aug 20 2022
%F A189911 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
%p A189911 A189911 := proc(n) local a,b,d; if n = 0 then 1 else
%p A189911 a := GAMMA(n-floor(n/2)); b := GAMMA(floor(n/2+3/2));
%p A189911 d := GAMMA(floor(n/2+1))^2; GAMMA(n+1)*(a*b+d)/(a*b*d) fi end: seq(A189911(n),n=0..32);
%p A189911 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
%t A189911 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 *)
%o A189911 (Sage)
%o A189911 def A189911():
%o A189911     r, n = 1, 1
%o A189911     while True:
%o A189911         yield r
%o A189911         h = n//2
%o A189911         r *= 2 if is_even(n) else (h+2)*(2*h+1)/(h+1)^2
%o A189911         n += 1
%o A189911 a = A189911(); [next(a) for i in range(16)]  # _Peter Luschny_, Oct 24 2013
%Y A189911 Cf. A037965, A097070, A056040, A212303.
%K A189911 nonn
%O A189911 0,2
%A A189911 _Peter Luschny_, May 01 2011