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

0,2

Links

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

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)

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}]

Prog

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

Crossrefs

Cf. A000346, A037965, A050157, A296770.

Sequence in context: A151224 A151225 A326984 * A270786 A151226 A151320

Adjacent sequences: A296768 A296769 A296770 * A296772 A296773 A296774

Keyword

nonn

Author

Peter Luschny, Dec 21 2017

Status

approved