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A034742 Dirichlet convolution of Moebius function mu(n) (A008683) with Catalan numbers (A000108). 1
1, 0, 1, 4, 13, 40, 131, 424, 1428, 4848, 16795, 58740, 208011, 742768, 2674425, 9694416, 35357669, 129643320, 477638699, 1767258324, 6564120287, 24466250224, 91482563639, 343059554440, 1289904147310, 4861946193440, 18367353070722, 69533550173100, 263747951750359 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

LINKS

Table of n, a(n) for n=1..29.

FORMULA

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

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

EXAMPLE

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

MATHEMATICA

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

PROG

(PARI) /* Dirichlet convolution of mu(n) with Catalan numbers: */

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

for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 04 2015

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

{a(n)=local(A=[1, 0]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(sum(n=1, #A, subst(x*Ser(A), x, (x-x^2 +x*O(x^#A))^n)))[#A]); A[n]}

for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 04 2015

CROSSREFS

Sequence in context: A171556 A227747 A094628 * A149424 A097112 A222270

Adjacent sequences:  A034739 A034740 A034741 * A034743 A034744 A034745

KEYWORD

nonn,easy

AUTHOR

Erich Friedman

STATUS

approved

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Last modified January 19 09:35 EST 2020. Contains 331048 sequences. (Running on oeis4.)