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A160479 The ZL(n) sequence of the Zeta and Lambda triangles A160474 and A160487 7
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, 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, 11, 113, 1, 1, 1, 1, 1, 1, 127, 2, 131 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

The rather strange ZL(n) sequence rules both the Zeta and Lambda triangles.

The Zeta triangle led to the first and the Lambda triangle to the second Maple algorithm.

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.

LINKS

Table of n, a(n) for n=3..66.

FORMULA

ZL(n) = (2*n-1) * (A217983(n-1)/A128060(n)) for n >= 3.

ZL(n) = ZETA(n, m)/(ZETA(n-1, m-1) - (n-1)^2 * ZETA(n-1, m)), see A160474.

ZL(n) = LAMBDA(n, m)/(LAMBDA(n-1, m-1) - (2*n-3)^2 * LAMBDA(n-1, m)), see A160487.

ZL(n) = A160476(n)/A160476(n-1)

MAPLE

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);

# End program 1 (program edited, Johannes W. Meijer, Oct 25 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)!) 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);

# End program 2 (program edited, Johannes W. Meijer, Sep 20 2012)

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);

# End program 3 (program added, Johannes W. Meijer, Oct 25 2012)

CROSSREFS

Cf. A160474 and A160487.

The cnf1(n, k) are the central factorial numbers A008955.

The cnf2(n, k) are the central factorial numbers A008956.

Sequence in context: A018990 A280882 A319599 * A085222 A085221 A128536

Adjacent sequences:  A160476 A160477 A160478 * A160480 A160481 A160482

KEYWORD

easy,nonn

AUTHOR

Johannes W. Meijer, May 24 2009

EXTENSIONS

Comments, formulas and third Maple program added by Johannes W. Meijer, Oct 25 2012.

STATUS

approved

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Last modified October 21 05:06 EDT 2018. Contains 316404 sequences. (Running on oeis4.)