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
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..1664
Programs
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Mathematica
Table[Total[Binomial[2#,#]&/@Divisors[n]],{n,30}] (* Harvey P. Dale, Aug 20 2022 *) -
PARI
a(n)=sumdiv(n,d,binomial(2*d,d))
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