A069945 Let M_k be the k X k matrix M_k(i,j)=1/binomial(i+n,j); then a(n)=1/det(M_(n+1)).

1, -6, -360, 252000, 2222640000, -258768639360000, -410299414270986240000, 9061429740221589431500800000, 2835046804394206618956825845760000000, -12733381268715468286016211650968992153600000000

Offset: 1

Author

Benoit Cloitre, Apr 27 2002

Keywords

• easy
• sign

Comments

If k>n+1 det(M_k)=0

Crossrefs

A005249, A056040.

Programs

Mathematica

a[n_] := (-1)^Quotient[n, 2]/(Det[HilbertMatrix[n]] n!); Array[a, 10] (* Jean-François Alcover, Jul 06 2019 *)

PARI

for(n=0,10,print1(1/matdet(matrix(n+1,n+1,i,j,1/binomial(i+n,j))),","))

Sage

def A069945(n): return (-1)^(n//2)*mul(binomial(i,i//2) for i in (1..2*n-1))
[A069945(i) for i in (1..11)] # Peter Luschny, Sep 18 2012

Formula

|a(n)| = det(M^(-1)), where M is an n X n matrix with M[i, j]=i/(i+j-1) (or j/(i+j-1)). |a(n)| = 1/det(HilbertMatrix(n))/n! = A005249(n)/n!. - Vladeta Jovovic, Jul 26 2003

|a(n)| = Product_{i=1..2n-1} binomial(i,floor(i/2)). - Peter Luschny, Sep 18 2012

|a(n)| = (Product_{i=1..2n-1} A056040(i))/n! = A163085(2*n-1)/n!. - Peter Luschny, Sep 18 2012