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A034731 Dirichlet convolution of b_n=1 with Catalan numbers.

%I A034731 #24 Jan 12 2021 15:34:31
%S A034731 1,2,3,7,15,46,133,436,1433,4878,16797,58837,208013,743034,2674457,
%T A034731 9695281,35357671,129646266,477638701,1767268073,6564120555,
%U A034731 24466283818,91482563641,343059672916,1289904147339,4861946609466
%N A034731 Dirichlet convolution of b_n=1 with Catalan numbers.
%C A034731 Also number of objects fixed by permutations A057509/A057510 (induced by shallow rotation of general parenthesizations/plane trees).
%H A034731 G. C. Greubel, <a href="/A034731/b034731.txt">Table of n, a(n) for n = 1..1000</a>
%F A034731 a(n) = Sum_{d divides n} C(d-1) where C() are the Catalan numbers (A000108).
%F A034731 a(n) ~ 4^(n-1) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Dec 05 2015
%F A034731 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
%F A034731 G.f.: Sum_{n>=1} (1 - sqrt(1 - 4*x^n))/2. - _Paul D. Hanna_, Jan 12 2021
%F A034731 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
%t A034731 a[n_] := DivisorSum[n, CatalanNumber[#-1]&]; Array[a, 26] (* _Jean-François Alcover_, Dec 05 2015 *)
%o A034731 (PARI) a(n) = sumdiv(n, d, binomial(2*(d-1),d-1)/d) \\ _Michel Marcus_, Jun 07 2013
%o A034731 (PARI) {a(n) = my(A = sum(m=1, n, (1 - sqrt(1 - 4*x^m +x*O(x^n)))/2 )); polcoeff(A, n)}
%o A034731 for(n=1, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jan 12 2021
%o A034731 (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)}
%o A034731 for(n=1, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jan 12 2021
%Y A034731 Occurs for first time in A073202 as row 16.
%Y A034731 Cf. A000108, A057509, A057510, A057546.
%K A034731 nonn
%O A034731 1,2
%A A034731 _Erich Friedman_
%E A034731 More comments from _Antti Karttunen_, Jan 03 2003