The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A034742 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A034742

Dirichlet convolution of Moebius function mu(n) (A008683) with Catalan numbers (A000108).
(history; published version)

#16 by Vaclav Kotesovec at Wed Sep 11 05:08:26 EDT 2019

STATUS

editing

approved

#15 by Vaclav Kotesovec at Wed Sep 11 05:08:20 EDT 2019

FORMULA

a(n) ~ 2^(2*n-2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 11 2019

STATUS

approved

editing

#14 by Vaclav Kotesovec at Tue Sep 10 12:17:54 EDT 2019

STATUS

editing

approved

#13 by Vaclav Kotesovec at Tue Sep 10 12:17:30 EDT 2019

MATHEMATICA

Table[Sum[MoebiusMu[n/d]*CatalanNumber[d-1], {d, Divisors[n]}], {n, 1, 30}] (* Vaclav Kotesovec, Sep 10 2019 *)

STATUS

approved

editing

#12 by Michael Somos at Sun Jan 04 23:42:49 EST 2015

STATUS

editing

approved

#11 by Michael Somos at Sun Jan 04 23:42:38 EST 2015

EXAMPLE

G.f. = x + x^3 + 4*x^4 + 13*x^5 + 40*x^6 + 131*x^7 + 424*x^8 + 1428*x^9 + ...

STATUS

proposed

editing

Discussion

Sun Jan 04

23:42

Michael Somos: Added more info.

#10 by Paul D. Hanna at Sun Jan 04 20:37:21 EST 2015

STATUS

editing

proposed

#9 by Paul D. Hanna at Sun Jan 04 20:37:16 EST 2015

FORMULA

G.f. A(x) satisfies: Sum_{n>=1} A((x-x^2)^n) = x+x^2. - Paul D. Hanna, Jan 04 2015

a(n) = Sum_{d|n} Moebius(n/d) * binomial(2*(d-1), d-1)/d. - Paul D. Hanna, Jan 04 2015

PROG

(PARI) /* G.f. satisfies: Sum_{n>=1} A((x-x^2)^n) = x+x^2. : */

STATUS

proposed

editing

#8 by Paul D. Hanna at Sun Jan 04 20:17:20 EST 2015

STATUS

editing

proposed

#7 by Paul D. Hanna at Sun Jan 04 20:17:17 EST 2015

PROG

({a(n) = sumdiv(n, d, moebius(n/d) * binomial(2*(d-1), d-1)/d)}

STATUS

proposed

editing