A108553 -id:A108553 - cp's OEIS frontend

A109000 Antidiagonal sums of square array A108998, in which row n equals the coordination sequence of B_n lattice.

1, 1, 3, 11, 37, 133, 479, 1719, 6121, 21609, 75675, 263171, 909899, 3130963, 10730891, 36639987, 124528283, 420319907, 1403656123, 4615627555, 14868713515, 46702912307, 142489152555, 421113970835, 1203581558011

Offset: 0

Paul D. Hanna, Jun 17 2005

nonn

Limit a(n+1)/a(n) ~ 3.3829757679..., real root of cubic (1+x+3*x^2-x^3). Compare to antidiagonal sums A108555 of square array A108553, in which row n equals the crystal ball sequence for D_n lattice.

Cf. A108998, A108999, A109001.

{a(n)=sum(k=0,n,sum(j=0,k, binomial(n-j-1,k-j)*(binomial(2*n-2*k+1,2*j)-2*(n-k)*binomial(n-k-1,j-1))))}

a(n) = Sum_{k=0..n} Sum_{j=0..k} C(n-j-1, k-j) * (C(2*n-2*k+1, 2*j)-2*(n-k)*C(n-k-1, j-1)).

A108557 Row sums of triangle A108556, in which row n equals the inverse binomial transform of the crystal ball sequence for D_n lattice.

Original entry on oeis.org

1, 3, 9, 63, 433, 2823, 17657, 107439, 642529, 3802167, 22357097, 130970271, 765564049, 4469342439, 26073165401, 152043343119, 886424978881, 5167271805207, 30119654732489, 175558462395135, 1023255914549617

Offset: 0

Paul D. Hanna, Jun 10 2005

Limit a(n+1)/a(n) = 3+sqrt(8) = 5.82842712...

Cf. A108553, A108556.

a(n)=local(A=vector(n+1,r,vector(n+1,c,if(r-1==0 || c-1==0,1,if(r-1==1,2*c-1, sum(j=0,c-1,binomial(r+c-j-2,c-j-1)*(binomial(2*r-2,2*j)-2*(r-1)*binomial(r-3,j-1)))))))); sum(k=0,n,polcoeff(subst(Ser(A[n+1]),x, x/(1+x))/(1+x),k))

G.f.: (1-9*x+19*x^2+33*x^3-80*x^4+12*x^5)/(1-12*x+46*x^2-60*x^3+9*x^4).

cp's OEIS Frontend

A109000 Antidiagonal sums of square array A108998, in which row n equals the coordination sequence of B_n lattice.

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