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A160474
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The Zeta triangle.
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13
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-1, 51, -10, -10594, 2961, -210, 356487, -115940, 12642, -420, -101141295, 35804857, -4751890, 254562, -4620, 48350824787, -18071509911, 2689347661, -180909586, 5471466, -60060
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OFFSET
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2,2
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COMMENTS
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The coefficients of the ZS1 matrix are defined by ZS1[2*m-1,n] = (2^(2*m-1))*int(y^(2*m-1)/(sinh(y))^(2*n), y=0..infinity)/factorial(2*m-1) for m = 1, 2, 3, .. and n = 1, 2, 3, .. under the condition that n <= (m-1).
This definition leads to ZS1[2*m-1,n=1] = 2*zeta(2*m-1), for m = 2, 3, .. , and the recurrence relation ZS1[2*m-1,n]:=(1/(2*n-1))*((2/(n-1))*ZS1[2*m-3,n-1]-(2*n-2)*ZS1[2*m-1,n-1]). As usual zeta(m) is the Riemann zeta function. These two formulas enable us to determine the values of the ZS[2*m-1,n] coefficients, with m all integers and n all positive integers, but not for all. If we choose, somewhat but not entirely arbitrarily, ZS1[1,n=1] = 2*gamma, with gamma the Euler-Mascheroni constant, we can determine them all.
The coefficients in the columns of the ZS1 matrix, for m = 1, 2, 3, .., and n = 2, 3, 4 .. , can be generated with the GH(z;n) polynomials for which we found the following general expression GH(z;n) = (h(n)*CFN1(z;n)*GH(z;n=1) + ZETA(z;n))/p(n).
The CFN1(z;n) polynomials depend on the central factorial numbers A008955.
The ZETA(z;n) are the Zeta polynomials which lead to the Zeta triangle.
The zero patterns of the Zeta polynomials resemble a UFO. These patterns resemble those of the Eta, Beta and Lambda polynomials, see A160464, A160480 and A160487.
The first Maple algorithm generates the coefficients of the Zeta triangle. The second Maple algorithm generates the ZS1[2*m-1,n] coefficients for m= 0, -1, -2, .. .
The M(n) sequence, see the second Maple algorithm, leads to Gould's sequence A001316 and a sequence that resembles the denominators in Taylor series for tan(x), i.e., A156769(n).
Some of our results are conjectures based on numerical evidence.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.
Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
Johannes W. Meijer, The zeros of the Eta, Zeta, Beta and Lambda polynomials, jpg and pdf, Mar 03 2013.
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FORMULA
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We discovered a remarkable relation between the Zeta triangle coefficients ZETA(n,m) = ZL(n)*(ZETA(n-1,m-1)-(n-1)^2*ZETA(n-1,m)) for n = 3, 4, ... and m = 2, 3, .... See A160475 for ZETA(n,m=1) and furthermore ZETA(n,n) = 0 for n = 2, 3, ....
We observe that the ZL(n) = A160479(n) sequence also rules the Lambda triangle A160487.
The generating functions GH(z;n) of the coefficients in the matrix columns are defined by
GH(z;n) = sum(ZS1[2*m-1,n]*z^(2*m-2), m=1..infinity), with n = 1, 2, 3, .... This definition, and our choice of ZS1[1,1] = 2*gamma, leads to GH(z;n=1) = (-Psi(1-z)-Psi(1+z)) with Psi(z) the digamma-function. Furthermore we discovered that GH(z;n) = GH(z;n-1)*(2*z^2/((2*n-1)*(n-1))-(2*n-2)/(2*n-1))+2*ZS1[ -1,n-1]/((2*n-1)*(n-1)) for n = 2, 3 , ..., with ZS1[ -1,n] = 2^(2*n-1)*A002195(n)/A002196(n) for n = 1, 2, ....
We found the following general expression for the GH(z;n) polynomials, for n = 2, 3, ...:
GH(z;n) = (h(n)*CFN1(z;n)*GH(z;n=1) + ZETA(z;n))/p(n) with
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EXAMPLE
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The first few rows of the triangle ZETA(n,m) with n=2,3,... and m=1,2,... are
[ -1],
[51, -10],
[ -10594, 2961, -210],
[356487, -115940, 12642, -420].
The first few ZETA(z;n) polynomials are
ZETA(z;n=2) = -1,
ZETA(z;n=3) = 51-10*z^2,
ZETA(z;n=4) = -10594 + 2961*z^2 - 210*z^4.
The first few CFN1(z;n) polynomials are
CFN1(z;n=2) = (z^2-1),
CFN1(z;n=3) = (z^4 - 5*z^2 + 4),
CFN1(z;n=4) = (z^6 - 14*z^4 + 49*z^2 - 36).
The first few generating functions GH(z;n) are
GH(z;n=2) = (6*(z^2-1)*GH(z;n=1) + (-1)) / 9,
GH(z;n=3) = (60*(z^4-5*z^2+4)*GH(z;n=1) + (51-10*z^2)) / 450,
GH(z;n=4) = (1260*(z^6-14*z^4+49*z^2-36)*GH(z;n=1) + (-10594+2961*z^2-210*z^4))/99225.
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MAPLE
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nmax:=7; 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 Zc(n) := (Omega(n)*2^(2*n-1))*2/((2*n+1)*(n)) end do: c(1) := denom(Zc(1)): for n from 2 to nmax do c(n) := lcm(c(n-1)*(n)*(2*n+1)/2, denom(Zc(n))); p(n) := c(n-1) end do: y(1):=Zc(1): for n from 1 to nmax-1 do y(n+1) := Zc(n+1) - ((2*n+2)/(2*n+3))*y(n) end do: for n from 1 to nmax do b(n) := 4^(-n)*(2*n+1)*n*denom(Omega(n)) end do: 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 1 to nmax-1 do ZL(n+2) := cm(n+1)/cm(n) end do: mmax := nmax: for n from 2 to nmax do ZETA(n, 1) := p(n)*y(n-1): ZETA(n, n) := 0 end do: for m from 2 to mmax do for n from m+1 to nmax do ZETA(n, m) := ZL(n)*(ZETA(n-1, m-1) - (n-1)^2* ZETA(n-1, m)) end do end do; seq(seq(ZETA(n, m), m=1..n-1), n=2..nmax);
nmax1 := 10; m := 1; ZS1row := 1-2*m; with(combinat): t1 := 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: mmax1 := nmax1: for m1 from 1 to mmax1 do M(m1-1) := 2^(2*m1-2)/((2*m1-1)!) end do: for m1 from 1 to mmax1 do ZS1[ -2*m1+1, 1] := 2*(-bernoulli(2*m1)/(2*m1)) od: for n from 2 to nmax1 do for m1 from 1 to mmax1-n+1 do ZS1[-2*m1+1, n] := M(n-1)*sum((-1)^(k1+1)*t1(n-1, k1-1) * ZS1[2*k1-2*n-2*m1+1, 1], k1 = 1..n) od: od: seq(ZS1[1-2*m, n], n = 1..nmax1-m+1);
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CROSSREFS
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A160475 equals the first left hand column.
A160476 equals the first right hand column and 6*h(n).
A001620 is the Euler-Mascheroni constant gamma.
The ZS1[ -1, n] and the Omega(n) coefficients lead to A002195 and A002196.
The CFN1(z, n) and the cfn1(n, k) lead to A008955.
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KEYWORD
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AUTHOR
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STATUS
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approved
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