A292347 - OEIS

%I #18 Dec 08 2017 22:56:37

%S 1,0,2,16,192,3008,58480,1360896,36931328,1145967616,40040976384,

%T 1556236513280,66610814414848,3113899625938944,157874306413611008,

%U 8629070019375726592,505841319779582607360,31659277087340088786944,2107162955059322401718272

%N Möbius function of absolute order.

%C (-1)^{n-1} a(n) is the Möbius function value mu(0,1) of the absolute order on the symmetric group S_n with a top element 1 adjoined.

%D R. Stanley, Enumerative Combinatorics, vol. 1, second ed., Cambridge University Press (2012), Exercise 3.159.

%H Alois P. Heinz, <a href="/A292347/b292347.txt">Table of n, a(n) for n = 1..367</a>

%F The exponential generating function for (-1)^{n-1} a(n) is exp(Sum_{p>=1} C(p-1) * x^p/p) = (-1+sqrt(1+4*x))*exp(-1+sqrt(1+4*x))/(2*x), where C(p-1) is a Catalan number.

%p a:= n-> n! * abs(coeff(series((sqrt(1+4*x)-1)*

%p         exp(sqrt(1+4*x)-1)/(2*x), x, n+3), x, n)):

%p seq(a(n), n=1..25);  # _Alois P. Heinz_, Dec 08 2017

%Y Cf. A000108, A008683.

%K nonn

%O 1,3

%A _Richard Stanley_, Dec 07 2017