A034742 - OEIS

A034742

Dirichlet convolution of Moebius function mu(n) (A008683) with Catalan numbers (A000108).

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

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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