A034731 Dirichlet convolution of b_n=1 with Catalan numbers.

1, 2, 3, 7, 15, 46, 133, 436, 1433, 4878, 16797, 58837, 208013, 743034, 2674457, 9695281, 35357671, 129646266, 477638701, 1767268073, 6564120555, 24466283818, 91482563641, 343059672916, 1289904147339, 4861946609466

Offset: 1

Author

Erich Friedman

Keywords

• nonn

Comments

Also number of objects fixed by permutations A057509/A057510 (induced by shallow rotation of general parenthesizations/plane trees).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Occurs for first time in A073202 as row 16.

Cf. A000108, A057509, A057510, A057546.

Programs

Mathematica

a[n_] := DivisorSum[n, CatalanNumber[#-1]&]; Array[a, 26] (* Jean-François Alcover, Dec 05 2015 *)

PARI

a(n) = sumdiv(n, d, binomial(2*(d-1),d-1)/d) \\ Michel Marcus, Jun 07 2013

PARI

{a(n) = my(A = sum(m=1, n, (1 - sqrt(1 - 4*x^m +x*O(x^n)))/2 )); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 12 2021

PARI

{a(n) = my(A = sum(m=1, n, binomial(2*m-2,m-1)/m * x^m/(1 - x^m +x*O(x^n)) )); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 12 2021

Formula

a(n) = Sum_{d divides n} C(d-1) where C() are the Catalan numbers (A000108).

a(n) ~ 4^(n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 05 2015

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(binomial(2*k-2,k-1)/k^2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018

G.f.: Sum_{n>=1} (1 - sqrt(1 - 4*x^n))/2. - Paul D. Hanna, Jan 12 2021

G.f.: Sum_{n>=1} A000108(n-1) * x^n/(1 - x^n) where A000108(n) = binomial(2*n,n)/(n+1). - Paul D. Hanna, Jan 12 2021

Extensions

More comments from Antti Karttunen, Jan 03 2003