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A034731
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Dirichlet convolution of b_n=1 with Catalan numbers.
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9
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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
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OFFSET
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1,2
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COMMENTS
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Also number of objects fixed by permutations A057509/A057510 (induced by shallow rotation of general parenthesizations/plane trees).
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LINKS
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FORMULA
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a(n) = Sum_{d divides n} C(d-1) where C() are the Catalan numbers (A000108).
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
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MATHEMATICA
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PROG
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(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)}
(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)}
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CROSSREFS
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Occurs for first time in A073202 as row 16.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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