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A349450
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Dirichlet inverse of right-shifted Catalan numbers [as when started from A000108(0): 1, 1, 2, 5, 14, 42, etc.]
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6
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1, -1, -2, -4, -14, -38, -132, -420, -1426, -4834, -16796, -58688, -208012, -742636, -2674384, -9693976, -35357670, -129641774, -477638700, -1767253368, -6564119892, -24466233428, -91482563640, -343059494120, -1289904147128, -4861945985428, -18367353066440, -69533549429280, -263747951750360, -1002242211282032
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OFFSET
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1,3
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LINKS
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FORMULA
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a(1) = 1; a(n) = -Sum_{d|n, d < n} A000108((n/d)-1) * a(d).
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} Catalan(k-1) * A(x^k). - Ilya Gutkovskiy, Feb 23 2022
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MATHEMATICA
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a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#] * CatalanNumber[n/# - 1] &, # < n &]; Array[a, 30] (* Amiram Eldar, Nov 22 2021 *)
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PROG
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(PARI)
A000108(n) = binomial(2*n, n)/(n+1);
memoA349450 = Map();
A349450(n) = if(1==n, 1, my(v); if(mapisdefined(memoA349450, n, &v), v, v = -sumdiv(n, d, if(d<n, A000108((n/d)-1)*A349450(d), 0)); mapput(memoA349450, n, v); (v)));
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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