This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A273055 #58 Mar 07 2025 01:29:02
%S A273055 1,7,51,393,3139,25653,212941,1787607,15134931,128996853,1105350729,
%T A273055 9513228123,82176836301,712070156203,6186675630819,53876592856681,
%U A273055 470139239360787,4109922421017093,35986168879543609,315544068167601787,2770417140954208377,24352194654450483759
%N A273055 a(n) = Sum_{k=0..n} binomial(2*k, k) * binomial(2*n+1, 2*k).
%C A273055 These are the central coefficients of the trinomial irregular triangle A027907 [Comtet, pp. 77-78, with references] for odd-indexed rows. For the central coefficients of the even-indexed rows see A082758. - _Wolfdieter Lang_, Apr 19 2018
%D A273055 L. Comtet, Advanced Combinatorics, Reidel, 1974., pp. 77-78. (In the integral formula on p. 77, a left bracket is missing for the cosine argument.)
%H A273055 Indranil Ghosh, <a href="/A273055/b273055.txt">Table of n, a(n) for n = 0..826</a>
%F A273055 a(n) = 4^(2*n+1)*JacobiP(2*n+1, -2*n-3/2, -2*n-3/2, -1/2).
%F A273055 a(n) = GegenbauerC(2*n+1, -2*n-1, -1/2).
%F A273055 a(n) = hypergeom([-n-1/2, -n], [1], 4).
%F A273055 a(n) = (2*n+1)!*( [x^(2*n+1)] exp(x)*BesselI(0,2*x) ).
%F A273055 a(n) = [x^(2*n+1)] (1-2*x-3*x^2)^(-1/2).
%F A273055 a(n) = [x^(2*n+1)] (1+x+x^2)^(2*n+1).
%F A273055 a(n) = ((4*n-1)*(20*n^2-10*n-3)*a(n-1)-9*(n-1)*(2*n-1)*(4*n+1)*a(n-2))/(n*(2*n+1)*(4*n-3)) for n>1.
%F A273055 a(n) = A002426(2*n+1).
%F A273055 a(n) ~ 3^(2*n + 3/2) / sqrt(8*Pi*n). - _Vaclav Kotesovec_, Feb 16 2017
%F A273055 From _Peter Bala_, Mar 16 2018: (Start)
%F A273055 a(n) = sqrt(-3)^(2*n+1)*P(2*n+1,1/sqrt(-3)), where P(n,x) is the Legendre polynomia1 of degree n.
%F A273055 a(n) = 1/C(2*n+1,n)*Sum_{k = 0..n} C(n,k)*C(n+k+1,k)* C(2*n+2*k+1,n+k+1)*(-3)^(n-k). Cf. A082758.
%F A273055 a(n) = (-3)^n*hypergeom([-n, n + 3/2], [1], 4/3).
%F A273055 (End)
%F A273055 From _Wolfdieter Lang_, Apr 19 2018: (Start)
%F A273055 a(n) = (2/Pi)*Integral_{phi = 0..Pi/2} (sin(3*phi)/sin(phi))^(2*n+1) dphi [Comtet, p. 77, q = 3, n = k -> 2*n+1] = (2/Pi)*Integral_{x=0..2} (x^2 - 1)^(2*n+1) / sqrt(4-x^2) dx (with x = 2*cos(phi)).
%F A273055 a(n) = 3^(2*n+1)*Sum_{k=0..2*n+1} binomial(2*n+1, k)*binomial(2*k, k)*(-1/3)^k = 3^(2*n+1)*hypergeometric([-(2*n+1), 1/2], [1], 4/3). See the version in Bala's formulas, also for the Legendre polynomial version. (End)
%F A273055 G.f.: sqrt( 2 / ((1-10*x+9*x^2)*(1-3*x+sqrt(1-10*x+9*x^2))) ). - _Seiichi Manyama_, Aug 25 2020
%p A273055 a := n -> GegenbauerC(2*n+1, -2*n-1, -1/2): seq(simplify(a(n)), n=0..21);
%t A273055 Table[Hypergeometric2F1[-n-1/2, -n, 1, 4], {n, 0, 21}]
%t A273055 Table[GegenbauerC[2 n + 1, -2 n - 1, -1/2], {n, 0, 21}] (* _Michael De Vlieger_, May 14 2016 *)
%o A273055 (Sage)
%o A273055 def a():
%o A273055 a, b, n = 1, 7, 2
%o A273055 yield a
%o A273055 while True:
%o A273055 yield b
%o A273055 a, b = b, ((4*n-1)*(20*n^2-10*n-3)*b-9*(n-1)*(2*n-1)*(4*n+1)*a)//(n*(2*n+1)*(4*n-3))
%o A273055 n += 1
%o A273055 A = a()
%o A273055 [next(A) for i in range(22)]
%o A273055 (PARI) a(n) = sum(k=0, n, binomial(2*k, k) * binomial(2*n+1, 2*k)); \\ _Michel Marcus_, Mar 01 2020
%o A273055 (PARI) N=40; x='x+O('x^N); Vec(sqrt(2/((1-10*x+9*x^2)*(1-3*x+sqrt(1-10*x+9*x^2))))) \\ _Seiichi Manyama_, Aug 25 2020
%Y A273055 Cf. A002426, A027907, A082758.
%K A273055 nonn,easy
%O A273055 0,2
%A A273055 _Peter Luschny_, May 14 2016