A220314 binomial(2n,n) - (2n)^pi(n), where pi(n) is the number of primes <= n.

1, 2, -16, 6, -748, -804, -34984, -52666, -56356, 24756, -4448200, -5258468, -298515176, -441773704, -573882480, -472661434, -50189743924, -69289028796, -4312446874696, -6415753471180, -9144394121976, -11944124661496, -913956731941456, -1320357856911588

Offset

1,2

Comments

a(n) is strictly positive for all n >= 202. In fact, Erdos and Ecklund-Eggleton proved more generally that binomial(k,n) > k^pi(n) if n >= 202 and k >= 2n. This theorem implies Sylvester's theorem. For the latter and references, see A213253.

Links

T. D. Noe, Table of n, a(n) for n = 1..500

Formula

a(n) = A000984(n) - (2n)^A000720(n).

Example

a(2) = binomial(4,2) - 4^pi(2) = 6 - 4 = 2.

Mathematica

Table[Binomial[2n, n] - (2n)^PrimePi[n], {n, 32}]

Crossrefs

Cf. A000720, A000984, A213253.

Sequence in context: A336833 A211367 A351585 * A302206 A110008 A296728

Adjacent sequences: A220311 A220312 A220313 * A220315 A220316 A220317

Keyword

sign

Author

Jonathan Sondow, Dec 10 2012

Status

approved