2,2

This intriguing sequence makes its appearance in the Zeta and Lambda triangles.

The first Maple algorithm is related to the Zeta triangle and the second to the Lambda triangle. Both generate the sequence of the first right hand column of these triangles.

a(n) = A160490(n)/(6*(2*n-2)!) for n = 2, 3, .. .

a(n) = A160478(n)*M(n) with M(n) = 2^(2*n-3)/(3*(2*n-1)!) for n=2, 3, .. .

M(n) = A048896(n-2)/(9*M1(n-1)) with M1(n) = (2*n+1)*A000265(n)*M1(n-1) for n = 2, 3, .. , and M1(1) = 1.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)

a(n+1)/a(n) = A160479(n+1)

(End)

restart; nmax:=21: jn:=nmax: im:=nmax: Omega(0):=1: for n from 1 to nmax do for j from 1 to jn do cfn1(1, j):=1 end do: for i from 2 to im do cfn1(i, 1):=0 end do: for j from 2 to jn do for i from 2 to im do cfn1(i, j):=cfn1(i-1, j-1)*(j-1)^2+cfn1(i, j-1) end do end do: Omega(n):= (sum((-1)^(k+n+1)*(bernoulli(2*k)/(2*k))*cfn1(n-k+1, n), k=1..n))/(2*n-1)! end do: for n from 1 to nmax do d(n):=(Omega(n)*2^(2*n-1)) end do: for n from 2 to nmax do Zc(n-1):= d(n-1)*2/((2*n-1)*(n-1)) end do: c(1):=denom(Zc(1)): for n from 1 to nmax-1 do c(n+1):= lcm(c(n)*(n+1)*(2*n+3)/2, denom(Zc(n+1))): p(n+1):=c(n) end do: for n from 2 to nmax do a(n):= p(n)*2^(2*n-3)/(3*factorial(2*n-1)) od: seq(a(n), n=2..nmax);

restart; nmax:=21; jn:=nmax+1: im:=nmax+1: for n from 1 to nmax do for i from 2 to im do cfn2(i, 1):=0 end do: for j from 1 to jn do cfn2(1, j):=1 end do: for j from 2 to jn do for i from 2 to im do cfn2(i, j):= cfn2(i, j-1) + cfn2(i-1, j-1)*(2*j-3)^2 end do end do: Delta(n-1):= sum((1-2^(2*k-1))* (-1)^(n+1)*(-bernoulli(2*k)/(2*k))*(-1)^(k+n)*cfn2(n-k+1, n), k=1..n) /(2*4^(n-1)*(2*n-1)!); LAMBDA(-2, n):= sum(2*(1-2^(2*k-1))*(-bernoulli(2*k)/ (2*k))*(-1)^(k+n)* cfn2(n+1-k, n), k=1..n)/ factorial(2*n-2) end do: Lcgz(2):=1/12: f(2):=1/12: for n from 3 to nmax do Lcgz(n):=LAMBDA(-2, n-1)/((2*n-2)*(2*n-3)): f(n):= Lcgz(n)-((2*n-3)/(2*n-2))*f(n-1) end do: for n from 1 to nmax do b(n):=denom(Lcgz(n+1)) end do: for n from 1 to nmax do b(n):=2*n*denom(Delta(n-1))/2^(2*n) end do: p(2):=b(1): for n from 2 to nmax do p(n+1):= lcm(p(n)*(2*n)*(2*n-1), b(n)) end do: for n from 2 to nmax do a(n):=p(n)/(6*factorial(2*n-2)) od: seq(a(n), n=2..nmax);

The Zeta and Lambda triangles are A160474 and A160487.

Cf. A160478 and A160490, A008955 and A008956, A048896 and A000265.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 2009: (Start)

Appears in A162446 (ZG1 matrix) and A162448 (LG1 matrix).

(End)

Sequence in context: A027014 A088746 A069863 this_sequence A067642 A052245 A052246

Adjacent sequences: A160473 A160474 A160475 this_sequence A160477 A160478 A160479

easy,nonn

Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009