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A034731 Dirichlet convolution of b_n=1 with Catalan numbers. 9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Also number of objects fixed by permutations A057509/A057510 (induced by shallow rotation of general parenthesizations/plane trees).
LINKS
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
MATHEMATICA
a[n_] := DivisorSum[n, CatalanNumber[#-1]&]; Array[a, 26] (* Jean-François Alcover, Dec 05 2015 *)
PROG
(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
CROSSREFS
Occurs for first time in A073202 as row 16.
Sequence in context: A368564 A213385 A045629 * A216435 A110888 A133736
KEYWORD
nonn
AUTHOR
EXTENSIONS
More comments from Antti Karttunen, Jan 03 2003
STATUS
approved

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Last modified March 19 01:34 EDT 2024. Contains 370952 sequences. (Running on oeis4.)