A158886 - OEIS

%I #12 Sep 08 2022 08:45:43

%S 1,1,-2,21,-504,21505,-1432080,137227545,-17893715840,3047775608241,

%T -657209398809600,175036741783305325,-56436686113876992000,

%U 21667473499647065000625,-9768377272589156352395264

%N a(n) = (n+1)^n * n! * C(1/(n+1), n).

%H G. C. Greubel, <a href="/A158886/b158886.txt">Table of n, a(n) for n = 0..230</a>

%F a(n) = Product_{k=0..n-1} (1 - k*(n+1)) for n>0 with a(0)=1.

%F a(n) = Coefficient of x^n/n! in (1 + (n+1)*x)^(1/(n+1)).

%F a(n) ~ (-1)^(n+1) * sqrt(2*Pi) * exp(1-n) * n^(2*n-3/2). - _Vaclav Kotesovec_, Jun 28 2015

%e a(1) = 1, a(2) = 1*(-2), a(3) = 1*(-3)*(-7), a(4) = 1*(-4)*(-9)*(-14).

%p seq( (n+1)^n*n!*binomial(1/(n+1), n), n=0..20); # _G. C. Greubel_, Mar 04 2020

%t Table[(n+1)^n n!Binomial[1/(n+1),n],{n,0,20}] (* _Harvey P. Dale_, Oct 17 2013 *)

%o (PARI) {a(n) = (n+1)^n * n! * binomial(1/(n+1),n)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n) = if(n==0,1, prod(k=0,n-1, 1 - k*(n+1) ))}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n) = n!*polcoeff( (1 + (n+1)*x +x*O(x^n))^(1/(n+1)),n)}

%o for(n=0,20,print1(a(n),", "))

%o (Magma) [1] cat [(&*[1-j*(n+1): j in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Mar 04 2020

%o (Sage) [(n+1)^n*factorial(n)*binomial(1/(n+1), n) for n in (0..20)] # _G. C. Greubel_, Mar 04 2020

%Y Cf. A158887.

%K sign

%O 0,3

%A _Paul D. Hanna_, May 01 2009