A072929 a(n) = Sum_{d dividing n} binomial(2d,d).

2, 8, 22, 78, 254, 952, 3434, 12948, 48642, 185016, 705434, 2705178, 10400602, 40120040, 155117794, 601093338, 2333606222, 9075184872, 35345263802, 137846713906, 538257877894, 2104099669160, 8233430727602, 32247606401148

Offset

1,1

Links

Seiichi Manyama, Table of n, a(n) for n = 1..1664

Formula

G.f.: Sum_{k>=1} C(2k, k)*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(binomial(2*k,k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 20 2018

a(n) ~ 4^n / sqrt(Pi*n). - Vaclav Kotesovec, May 21 2018

a(n) = Sum_{k=1..n} C(2*gcd(n,k),gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} C(2*n/gcd(n,k),n/gcd(n,k))/phi(n/gcd(n,k)) where phi = A000010. - Richard L. Ollerton, May 19 2021

Mathematica

Table[Total[Binomial[2#, #]&/@Divisors[n]], {n, 30}] (* Harvey P. Dale, Aug 20 2022 *)

Prog

(PARI) a(n)=sumdiv(n, d, binomial(2*d, d))

Crossrefs

Cf. A000984.

Cf. A000010.

Sequence in context: A106053 A121135 A183410 * A328140 A053958 A300370

Adjacent sequences: A072926 A072927 A072928 * A072930 A072931 A072932

Keyword

easy,nonn

Author

Benoit Cloitre, Aug 13 2002

Status

approved