A160476 The first right hand column of the Zeta and Lambda triangles

1, 10, 210, 420, 4620, 60060, 60060, 2042040, 116396280, 581981400, 13385572200, 13385572200, 13385572200, 388181593800, 12033629407800, 24067258815600, 24067258815600, 890488576177200, 890488576177200

Offset

2,2

Comments

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.

Links

Table of n, a(n) for n=2..20.

Formula

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.

a(n+1)/a(n) = A160479(n+1) [Johannes W. Meijer, Oct 07 2009]

Maple

nmax := 20; with(combinat): cfn1 := proc(n, k): sum((-1)^j*stirling1(n+1, n+1-k+j) * stirling1(n+1, n+1-k-j), j=-k..k) end proc: Omega(0) := 1: for n from 1 to nmax do Omega(n) := (sum((-1)^(k1+n+1)*(bernoulli(2*k1)/(2*k1))*cfn1(n-1, n-k1), k1=1..n))/(2*n-1)! end do: for n from 1 to nmax do d(n) := 2^(2*n-1)*Omega(n) 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 a1(n) := p(n)*2^(2*n-3)/(3*factorial(2*n-1)) od: seq(a1(n), n=2..nmax);
# End first program (program edited, Johannes W. Meijer, Sep 20 2012)
nmax1 := nmax: for n from 0 to nmax1 do cfn2(n, 0) := 1: cfn2(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax1 do for k from 1 to n-1 do cfn2(n, k) := (2*n-1)^2*cfn2(n-1, k-1) + cfn2(n-1, k) od: od: for n from 1 to nmax1 do Delta(n-1) := sum((1-2^(2*k1-1))* (-1)^(n+1)*(-bernoulli(2*k1)/(2*k1))*(-1)^(k1+n)*cfn2(n-1, n-k1), k1=1..n) /(2*4^(n-1)*(2*n-1)!); LAMBDA(-2, n) := sum(2*(1-2^(2*k1-1))*(-bernoulli(2*k1)/ (2*k1))*(-1)^(k1+n)* cfn2(n-1, n-k1), k1=1..n)/ factorial(2*n-2) end do: Lcgz(2) := 1/12: f(2) := 1/12: for n from 3 to nmax1 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 nmax1 do b(n) := denom(Lcgz(n+1)) end do: for n from 1 to nmax1 do b(n) := 2*n*denom(Delta(n-1))/2^(2*n) end do: p(2) := b(1): for n from 2 to nmax1 do p(n+1) := lcm(p(n)*(2*n)*(2*n-1), b(n)) end do: for n from 2 to nmax1 do a2(n) := p(n)/(6*factorial(2*n-2)) od: seq(a2(n), n=2..nmax1);
# End second program (program edited, Johannes W. Meijer, Sep 20 2012)

Crossrefs

The Zeta and Lambda triangles are A160474 and A160487.

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

Appears in A162446 (ZG1 matrix) and A162448 (LG1 matrix) [Johannes W. Meijer, Jul 06 2009]

Sequence in context: A335673 A215555 A069863 * A067642 A334537 A052245

Adjacent sequences: A160473 A160474 A160475 * A160477 A160478 A160479

Keyword

easy,nonn

Author

Johannes W. Meijer, May 24 2009

Status

approved