A136046 Bisection of A138543.

1, 3, 26, 345, 5754, 110586, 2341548, 53208441, 1276027610, 31930139670, 826963069140, 22035414489270, 601361536493340, 16749316314679500, 474777481850283240, 13665774112508864385, 398682239947705700730, 11770712453752716494910, 351240103372615793928900, 10581780543413346794758770

Offset: 0

Author

N. J. A. Sloane, Mar 25 2008

Keywords

• nonn

Crossrefs

Cf. A001405, A005558, A138543.

Programs

Mathematica

Q2[n_] := Quotient[n, 2];
A136046[n_] := Sum[(-1)^k*Binomial[2n, k] CatalanNumber[Q2[k+1]] CatalanNumber[Q2[k]](2*Q2[k]+1) Binomial[2n-k, Q2[2*n-k]], {k, 0, 2n + 1}];
Array[A136046, 20, 0]  (* After Mélika Tebni,  Peter Luschny, Jun 30 2025 *)

Python

from math import comb as C
def CN(n): return (C(2*n, n)//(n+1)) # Catalan numbers
def a(n): return sum((-1)**k*C(2*n, k)*CN((k+1)//2)*CN(k//2)*(2*(k//2)+1)*C(2*n-k, (2*n-k)//2) for k in range(2*n+1))  # Mélika Tebni, Jun 30 2025

Formula

a(n) = Sum_{k=0..2*n} (-1)^k*binomial(2*n, k)*A005558(k)*A001405(2*n-k). - Mélika Tebni, Jun 30 2025

Extensions

a(16)-a(19) from Mélika Tebni, Jun 30 2025