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A034731 Dirichlet convolution of b_n=1 with Catalan numbers. 7
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

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

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

MATHEMATICA

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

PROG

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

CROSSREFS

Occurs for first time in A073202 as row 16.

Cf. A000108, A057509, A057510, A057546.

Sequence in context: A213385 A161746 A045629 * A216435 A110888 A133736

Adjacent sequences:  A034728 A034729 A034730 * A034732 A034733 A034734

KEYWORD

nonn

AUTHOR

Erich Friedman

EXTENSIONS

More comments from Antti Karttunen, Jan 03 2003

STATUS

approved

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Last modified November 20 05:07 EST 2019. Contains 329323 sequences. (Running on oeis4.)