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

%I A160476 #9 Jul 22 2025 06:24:54
%S A160476 1,10,210,420,4620,60060,60060,2042040,116396280,581981400,
%T A160476 13385572200,13385572200,13385572200,388181593800,12033629407800,
%U A160476 24067258815600,24067258815600,890488576177200,890488576177200
%N A160476 The first right hand column of the Zeta and Lambda triangles.
%C A160476 This intriguing sequence makes its appearance in the Zeta and Lambda triangles.
%C A160476 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.
%F A160476 a(n) = A160490(n)/(6*(2*n-2)!) for n = 2, 3, .. .
%F A160476 a(n) = A160478(n)*M(n) with M(n) = 2^(2*n-3)/(3*(2*n-1)!) for n=2, 3, .. .
%F A160476 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.
%F A160476 a(n+1)/a(n) = A160479(n+1) [_Johannes W. Meijer_, Oct 07 2009]
%p A160476 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);
%p A160476 # End first program (program edited, _Johannes W. Meijer_, Sep 20 2012)
%p A160476 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);
%p A160476 # End second program (program edited, _Johannes W. Meijer_, Sep 20 2012)
%Y A160476 The Zeta and Lambda triangles are A160474 and A160487.
%Y A160476 Cf. A160478 and A160490, A008955 and A008956, A048896 and A000265.
%Y A160476 Appears in A162446 (ZG1 matrix) and A162448 (LG1 matrix) [_Johannes W. Meijer_, Jul 06 2009]
%K A160476 easy,nonn
%O A160476 2,2
%A A160476 _Johannes W. Meijer_, May 24 2009