A277924 a(n) = Sum_{i=0..n+1} binomial(2*n,n-i+1).

1, 4, 15, 57, 219, 848, 3302, 12911, 50643, 199140, 784626, 3096514, 12236830, 48412432, 191718188, 759852347, 3013746563, 11960699132, 47494802618, 188689585982, 749961486698, 2981943800192, 11860758904148, 47191458566582

Offset

0,2

Comments

a(n) = A124234(2*n+1,n) is diagonal of triangle A124234.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1661

Formula

G.f.: x^2*(1 - 2*x - sqrt(1 - 4*x))/(sqrt(1 - 4*x)*(4*x^3 - 13*x^2 + 7*x - 1) - 20*x^3 + 25*x^2 - 9*x + 1) - 1/x = (-1 + 5*x + (1 - x)*sqrt(1 - 4*x))/(2*x*(1 - 4*x)).

a(n) = 2^(2*n-1) - 4^n*(n-1/2)!/(sqrt(Pi)*(n+1)!) + 3*2^(2*n-1)*(n-1/2)!/(sqrt(Pi)*n!). - Bruno Berselli, Nov 04 2016

a(n) = ((3*n+1)*Catalan(n)+4^n)/2. - Peter Luschny, Nov 04 2016

Maple

a:= proc(n) option remember; `if`(n<3,
      [1, 4, 15][n+1], ((2*(4*n^2-7*n-1))*a(n-1)
      -(8*(n-1))*(2*n-3)*a(n-2))/((n-2)*(n+1)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 04 2016

Mathematica

Table[((3n + 1) CatalanNumber[n] + 4^n)/2, {n, 0, 23}] (* Peter Luschny, Nov 04 2016 *)

Prog

(Maxima) makelist(sum(binomial(2*n, n-i+1), i, 0, n+1), n, 0, 30);
(PARI) a(n) = sum(k=0, n+1, binomial(2*n, n-k+1)); \\ Michel Marcus, Nov 04 2016
(Sage)
def a():
    c, f, b, n = 1, 1, 1, 1
    while True:
        yield (c*f + b)//2
        c = (4*n - 2)*c//(n + 1)
        b <<= 2
        f += 3
        n += 1
A277924 = a()
print([next(A277924) for _ in range(23)]) # Peter Luschny, Nov 04 2016

Crossrefs

Cf. A000108, A124234.

Sequence in context: A242781 A346195 A371854 * A371777 A095930 A026850

Adjacent sequences: A277921 A277922 A277923 * A277925 A277926 A277927

Keyword

nonn

Author

Vladimir Kruchinin, Nov 04 2016

Status

approved