A273055 a(n) = Sum_{k=0..n} binomial(2*k, k) * binomial(2*n+1, 2*k).

1, 7, 51, 393, 3139, 25653, 212941, 1787607, 15134931, 128996853, 1105350729, 9513228123, 82176836301, 712070156203, 6186675630819, 53876592856681, 470139239360787, 4109922421017093, 35986168879543609, 315544068167601787, 2770417140954208377, 24352194654450483759

Offset

0,2

Comments

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

References

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.)

Links

Indranil Ghosh, Table of n, a(n) for n = 0..826

Formula

a(n) = 4^(2*n+1)*JacobiP(2*n+1, -2*n-3/2, -2*n-3/2, -1/2).

a(n) = GegenbauerC(2*n+1, -2*n-1, -1/2).

a(n) = hypergeom([-n-1/2, -n], [1], 4).

a(n) = (2*n+1)!*( [x^(2*n+1)] exp(x)*BesselI(0,2*x) ).

a(n) = [x^(2*n+1)] (1-2*x-3*x^2)^(-1/2).

a(n) = [x^(2*n+1)] (1+x+x^2)^(2*n+1).

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.

a(n) = A002426(2*n+1).

a(n) ~ 3^(2*n + 3/2) / sqrt(8*Pi*n). - Vaclav Kotesovec, Feb 16 2017

From Peter Bala, Mar 16 2018: (Start)

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.

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.

a(n) = (-3)^n*hypergeom([-n, n + 3/2], [1], 4/3).

(End)

From Wolfdieter Lang, Apr 19 2018: (Start)

a(n) = (2/Pi)*Integral_{phi = 0..Pi/2} (sin(3*phi))/sin(phi))^(2*n+1) [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) (with x = 2*cos(phi).

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)

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

Maple

a := n -> GegenbauerC(2*n+1, -2*n-1, -1/2): seq(simplify(a(n)), n=0..21);

Mathematica

Table[Hypergeometric2F1[-n-1/2, -n, 1, 4], {n, 0, 21}]
Table[GegenbauerC[2 n + 1, -2 n - 1, -1/2], {n, 0, 21}] (* Michael De Vlieger, May 14 2016 *)

Prog

(Sage)
def a():
    a, b, n = 1, 7, 2
    yield a
    while True:
        yield b
        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))
        n += 1
A = a()
[next(A) for i in range(22)]
(PARI) a(n) = sum(k=0, n, binomial(2*k, k) * binomial(2*n+1, 2*k)); \\ Michel Marcus, Mar 01 2020
(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

Crossrefs

Cf. A002426, A027907, A082758.

Sequence in context: A104454 A332936 A222849 * A019472 A219306 A246572

Adjacent sequences: A273052 A273053 A273054 * A273056 A273057 A273058

Keyword

nonn,easy

Author

Peter Luschny, May 14 2016

Status

approved