cp's OEIS Frontend

A276666 a(n) = (n-1)*Catalan(n).

%I A276666 #42 Mar 22 2022 05:53:29
%S A276666 -1,0,2,10,42,168,660,2574,10010,38896,151164,587860,2288132,8914800,
%T A276666 34767720,135727830,530365050,2074316640,8119857900,31810737420,
%U A276666 124718287980,489325340400,1921133836440,7547311500300,29667795388452,116686713634848,459183826803800
%N A276666 a(n) = (n-1)*Catalan(n).
%H A276666 G. C. Greubel, <a href="/A276666/b276666.txt">Table of n, a(n) for n = 0..1000</a>
%F A276666 a(n) = [x^n] (1-3*x)/(x*sqrt(1-4*x))-1/x.
%F A276666 a(n) = 4^n*(n-1)*hypergeom([3/2, -n], [2], 1).
%F A276666 a(n) = 4^n*(n-1)*JacobiP(n,1,-1/2-n,-1)/(n+1).
%F A276666 a(n) = (2*n)! [x^(2^n)]( BesselI(2,2*x) - (1+1/x)*BesselI(1,2*x) ).
%F A276666 a(n) = binomial(2*n,n) - 2*Catalan(n). (See _Geoffrey Critzer_'s formula in A024483).
%F A276666 a(n) = A056040(2*n) - 2*A057977(2*n).
%F A276666 a(n) = A056040(2*n)*(1-2/(n+1)) = (n^2-1)*(2*n)!/(n+1)!^2.
%F A276666 a(n) = A232500(2*n).
%F A276666 a(n) = a(n-1)*2*(n-1)*(2*n-1)/((n-2)*(n+1)) for n > 2. - _Chai Wah Wu_, Sep 12 2016
%F A276666 a(n) = A024483(n+1) for n>0. - _R. J. Mathar_, Sep 13 2016
%F A276666 a(n) = A000984(n+1)-3*A000984(n). - _Ezhilarasu Velayutham_, Aug 27 2019
%F A276666 From _Amiram Eldar_, Mar 22 2022: (Start)
%F A276666 Sum_{n>=2} 1/a(n) = 5/6 - Pi/(9*sqrt(3)).
%F A276666 Sum_{n>=2} (-1)^n/a(n) = 26*sqrt(5)*log(phi)/25 - 7/10, where phi is the golden ratio (A001622). (End)
%p A276666 f := (1-3*x)/(x*sqrt(1-4*x))-1/x:
%p A276666 series(f,x,29): seq(coeff(%,x,n), n=0..26);
%p A276666 A276666 := n -> (n^2-1)*(2*n)!/(n+1)!^2:
%p A276666 seq(A276666(n), n=0..26);
%t A276666 Table[(n - 1) CatalanNumber[n], {n, 0, 30}] (* _Vincenzo Librandi_, Sep 13 2016 *)
%o A276666 (Sage)
%o A276666 A276666 = lambda n: (n - 1) * catalan_number(n)
%o A276666 [A276666(n) for n in range(27)]
%o A276666 (Magma) [(n-1)*Catalan(n): n in [0..30]]; // _Vincenzo Librandi_, Sep 13 2016
%o A276666 (PARI) a(n) = if(n==0,-1, 2*binomial(2*n-1, n+1)); \\ _G. C. Greubel_, Aug 29 2019
%o A276666 (GAP) Concatenation([-1], List([1..30], n-> 2*Binomial(2*n-1, n+1))); # _G. C. Greubel_, Aug 29 2019
%Y A276666 A024483 is a variant of this sequence.
%Y A276666 Cf. A000108, A000984, A001622, A051631, A056040, A057977, A232500.
%K A276666 sign
%O A276666 0,3
%A A276666 _Peter Luschny_, Sep 12 2016