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A160479 The ZL(n) sequence of the Zeta and Lambda triangles A160474 and A160487.

%I A160479 #18 Apr 15 2021 22:55:45
%S A160479 10,21,2,11,13,1,34,57,5,23,1,1,29,31,2,1,37,1,41,301,1,47,1,1,53,3,1,
%T A160479 59,61,1,2,67,1,71,73,1,1,79,1,83,1,1,89,1,1,1,97,1,505,103,1,107,109,
%U A160479 11,113,1,1,1,1,1,1,127,2,131
%N A160479 The ZL(n) sequence of the Zeta and Lambda triangles A160474 and A160487.
%C A160479 The rather strange ZL(n) sequence rules both the Zeta and Lambda triangles.
%C A160479 The Zeta triangle led to the first and the Lambda triangle to the second Maple algorithm.
%C A160479 The first ZL(n) formula is a conjecture. This formula links the ZL(n) to the prime numbers A000040; see A217983, A128060, A130290 and the third Maple program.
%F A160479 ZL(n) = (2*n-1) * (A217983(n-1)/A128060(n)) for n >= 3.
%F A160479 ZL(n) = ZETA(n, m)/(ZETA(n-1, m-1) - (n-1)^2 * ZETA(n-1, m)), see A160474.
%F A160479 ZL(n) = LAMBDA(n, m)/(LAMBDA(n-1, m-1) - (2*n-3)^2 * LAMBDA(n-1, m)), see A160487.
%F A160479 ZL(n) = A160476(n)/A160476(n-1).
%p A160479 nmax := 65; for n from 0 to nmax do cfn1(n, 0):=1: cfn1(n, n):=(n!)^2 end do: for n from 1 to nmax do for k from 1 to n-1 do cfn1(n, k) := cfn1(n-1, k-1)*n^2 + cfn1(n-1, k) end do: end do: 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 1 to nmax do b(n) := 4^(-n)*(2*n+1)*n*denom(Omega(n)) end do: c(1) := b(1): for n from 1 to nmax-1 do c(n+1) := lcm(c(n)*(n+1)*(2*n+3)/2, b(n+1)) end do: for n from 1 to nmax do cm(n) := c(n)*(1/6)* 4^n/(2*n+1)! end do: for n from 3 to nmax+1 do ZL(n):=cm(n-1)/cm(n-2) end do: seq(ZL(n), n=3..nmax+1);
%p A160479 # End program 1 (program edited by _Johannes W. Meijer_, Oct 25 2012)
%p A160479 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)!) end do: for n from 1 to nmax1 do b(n) := (2*n)*(2*n-1)*denom(Delta(n-1))/ (2^(2*n)*(2*n-1)) end do: c(1) := b(1): for n from 1 to nmax1-1 do c(n+1) := lcm(c(n)*(2*n+2)* (2*n+1), b(n+1)) end do: for n from 1 to nmax1 do cm(n) := c(n)/(6*(2*n)!) end do: for n from 3 to nmax1+1 do ZL(n) := cm(n-1)/cm(n-2) end do: seq(ZL(n), n=3..nmax1+1);
%p A160479 # End program 2 (program edited by _Johannes W. Meijer_, Sep 20 2012)
%p A160479 nmax2 := nmax: A000040 := proc(n): ithprime(n) end: A130290 := proc(n): if n =1 then 1 else (A000040(n)-1)/2 fi: end: A128060 := proc(n) local n1: n1:=2*n-1: if type(n1, prime) then A128060(n) := 1 else A128060(n) := n1 fi: end: for n from 1 to nmax2 do A217983(n) := 1 od: for n from 1 to nmax2 do for n1 from 1 to floor(log[A000040(n)](nmax2)) do A217983(A130290(n) * A000040(n)^n1) := A000040(n) od: od: ZL := proc(n): (2*n-1)*(A217983(n-1)/A128060(n)) end: seq(ZL(n), n=3..nmax2+1);
%p A160479 # End program 3 (program added by _Johannes W. Meijer_, Oct 25 2012)
%Y A160479 Cf. A160474 and A160487.
%Y A160479 The cnf1(n, k) are the central factorial numbers A008955.
%Y A160479 The cnf2(n, k) are the central factorial numbers A008956.
%K A160479 easy,nonn,uned
%O A160479 3,1
%A A160479 _Johannes W. Meijer_, May 24 2009
%E A160479 Comments, formulas and third Maple program added by _Johannes W. Meijer_, Oct 25 2012