A330801 a(n) = A080247(2*n, n), the central values of the Big-Schröder triangle.

1, 4, 30, 264, 2490, 24396, 244790, 2496528, 25763058, 268243860, 2812481870, 29653804824, 314097641130, 3339741725404, 35626286189670, 381098437754912, 4086504567333858, 43912100376527652, 472743964145437310, 5097853987059017000, 55054474579787825562

Offset: 0

Author

Peter Luschny, Jan 02 2020

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..950

Crossrefs

Row sums of A330798.

Cf. A080247.

Programs

Magma

A330801:= func< n | ((n+1)/(2*n+1))*(&+[Binomial(2*n+1, n+j+1)*Binomial(2*n+j, j): j in [0..n]]) >;
[A330801(n): n in [0..40]]; // G. C. Greubel, May 03 2023

Maple

a := n -> ((n+1)/(2*n+1))*binomial(2*n+1, n+1)*hypergeom([-n, 2*n+1], [n+2], -1):
seq(simplify(a(n)), n=0..20);
# Alternative:
alias(C = binomial):
a := n -> ((n+1)/(2*n+1))*add(C(2*n+1, n+j+1)*C(2*n+j, j), j=0..n):
seq(a(n), n=0..20);

Mathematica

a[n_]:= (1/Sqrt[Pi]) 4^n (1 + n) Gamma[1/2 + n] Hypergeometric2F1Regularized[-n, 1 + 2 n, 2 + n, -1]; Table[a[n], {n, 0, 20}]

SageMath

def A330801(n) -> int:
    s = sum( binomial(2 * n + 1, n + j + 1) * binomial(2 * n + j, j)
        for j in range(n + 1) )
    return (s * (n + 1)) // (2 * n + 1)
print([A330801(n) for n in range(41)])  # G. C. Greubel, May 03 2023

Formula

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

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

D-finite with recurrence 2*n*(2*n+1)*(7*n-13)*a(n) - (382*n^3 -983*n^2 +533*n -40)*a(n-1) + (n-2)*(786*n^2 -3290*n +3315)*a(n-2) + (2*n-5)*(37*n-39)*(n-3)*a(n-3) = 0. - R. J. Mathar, Jul 27 2022

a(n) ~ phi^(5*n + 1) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 03 2023