A237670 Inverse Moebius transform of Catalan numbers.

1, 3, 6, 17, 43, 140, 430, 1447, 4868, 16841, 58787, 208166, 742901, 2674872, 9694893, 35359117, 129644791, 477643702, 1767263191, 6564137275, 24466267455, 91482622429, 343059613651, 1289904356920, 4861946401495, 18367353815055, 69533550920872

Offset

1,2

Links

Bruno Berselli, Table of n, a(n) for n = 1..200

Formula

a(n) = Sum_{d|n} binomial(2*d,d)/(d+1).

G.f.: Sum_{k>=1} (1-2*x^k-sqrt(1-4*x^k))/(2*x^k).

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

Mathematica

Table[Sum[Binomial[2 d, d]/(d + 1), {d, Divisors[n]}], {n, 1, 100}]

Prog

(Maxima) a(n) := lsum(binomial(2*d, d)/(d+1), d, listify(divisors(n)));
makelist(a(n), n, 1, 40);
(Magma) [&+[Binomial(2*d, d)/(d+1): d in Divisors(n)]: n in [1..40]]; // Bruno Berselli, Feb 11 2014
(PARI) a(n) = sumdiv(n, d, binomial(2*d, d)/(d+1)); \\ Michel Marcus, May 20 2018

Crossrefs

Cf. A000108.

Sequence in context: A129905 A143363 A216878 * A321227 A006081 A099511

Adjacent sequences: A237667 A237668 A237669 * A237671 A237672 A237673

Keyword

nonn

Author

Emanuele Munarini, Feb 11 2014

Status

approved