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

0,2

Links

Table of n, a(n) for n=0..19.

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

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 *)

Prog

(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

Crossrefs

Cf. A001405, A005558, A138543.

Sequence in context: A366315 A384024 A328269 * A206404 A373425 A262301

Adjacent sequences: A136043 A136044 A136045 * A136047 A136048 A136049

Keyword

nonn

Author

N. J. A. Sloane, Mar 25 2008

Extensions

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

Status

approved