A296771 Row sums of A050157.

1, 3, 13, 58, 257, 1126, 4882, 20980, 89497, 379438, 1600406, 6720748, 28117498, 117254268, 487589572, 2022568168, 8371423177, 34581780478, 142605399982, 587138954428, 2413944555742, 9911778919348, 40650232625212, 166534680737368, 681576405563722

Offset: 0

Author

Peter Luschny, Dec 21 2017

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..1657

Crossrefs

Cf. A000346, A037965, A050157, A296770.

Programs

Maple

A296771 := n -> add(binomial(2*n, n) - binomial(2*n, n+k+1), k=0..n):
seq(A296771(n), n=0..24);

Mathematica

a[n_] := 4^n ((n - 1/2)! (2 n + 3)/(2 Sqrt[Pi] n!) - 1/2);
Table[a[n], {n, 0, 24}]

PARI

a(n) = sum(k=0, n, binomial(2*n, n) - binomial(2*n, n+k+1)) \\ Iain Fox, Dec 21 2017

Formula

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

a(n) = 2^(2*n-1)*(((n-1/2)!*(2*n+3))/(sqrt(Pi)*n!) - 1).

a(n) ~ 4^n*(sqrt(n/Pi) - 1/2).

a(n) = A037965(n+1) - A000346(n-1) for n >= 1.

From Robert Israel, Dec 21 2017: (Start)

a(n) = (n+3/2)*binomial(2*n,n) - 2^(2*n-1).

G.f.: (3/2-4*x)*(1-4*x)^(-3/2) - (1/2)*(1-4*x)^(-1).

64*(n+1)*(2*n+1)*a(n)-8*(2*n+3)*(5*n+4)*a(n+1)+2*(n+2)*(8*n+11)*a(n+2)-(n+3)*(n+2)*a(n+3)=0. (End)