A234040 a(n) = binomial(2*(n+1),n) * gcd(n,2)/(2*(n+1)).

1, 1, 5, 7, 42, 66, 429, 715, 4862, 8398, 58786, 104006, 742900, 1337220, 9694845, 17678835, 129644790, 238819350, 1767263190, 3282060210, 24466267020, 45741281820, 343059613650, 644952073662, 4861946401452, 9183676536076, 69533550916004, 131873975875180, 1002242216651368

Offset

0,3

Comments

This gives the next-to-central entries of the even-indexed rows of the triangle A107711.

For the central entries (of the even-numbered rows) see A001700.

This sequence is composed of the bisection sequences A024492 (even part) and A065097 (odd part).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = binomial(2*(n+1),n)*gcd(n,2)/(2*(n+1)) for n >= 0.

a(n) = A107711(2*(n+1), n) for n >= 0.

G.f.: (3*c(x)- c(-x)-2)/(4*x) =(4*(1-x) - 3*sqrt(1-4*x) - sqrt(1+4*x))/(8*x^2), with c(x) the o.g.f. of the Catalan numbers A000108. See the bisection comment above.

a(n) ~ gcd(n, 2) * 2^(2*n+1) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 25 2025

Mathematica

Table[Binomial[2 (n + 1), n] GCD[n, 2]/(2 (n + 1)), {n, 0, 40}] (* Vincenzo Librandi, Feb 25 2014 *)

Prog

(Magma) [Binomial(2*(n+1), n)*Gcd(n, 2)/(2*(n+1)): n in [0..30]]; // Vincenzo Librandi, Feb 25 2014

Crossrefs

Cf. A000108, A024492, A065097, A107711.

Sequence in context: A117751 A093526 A098512 * A393134 A292010 A379639

Adjacent sequences: A234037 A234038 A234039 * A234041 A234042 A234043

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 23 2014

Extensions

a(26) from Vincenzo Librandi, Feb 25 2014

Status

approved