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%I A160464
%S A160464 1,11,2,114,29,2,3963,1156,122,4,104745,32863,4206,222,4,
%T A160464 3926745,1287813,184279,12198,366,4,198491580,67029582,10317484,
%U A160464 781981,30132,562,4
%V A160464 -1,-11,2,-114,29,-2,-3963,1156,-122,4,-104745,32863,-4206,222,-4,
%W A160464 -3926745,1287813,-184279,12198,-366,4,-198491580,67029582,-10317484,
%X A160464 781981,-30132,562,-4
%N A160464 The Eta triangle
%C A160464 The ES1 matrix coefficients are defined by ES1[2*m-1,n] = 2^(2*m-1) * 
               int(y^(2*m-1)/(cosh(y))^(2*n),y=0..infinity)/(2*m-1)! for m = 1, 
               2, 3, .. and n = 1, 2, 3 .. .
%C A160464 This definition leads to ES1[2*m-1,n=1] = 2*eta(2*m-1) and the recurrence 
               relation ES1[2*m-1,n] = ((2*n-2)/(2*n-1))*(ES1[2*m-1,n-1] - ES1[2*m-3,
               n-1]/(n-1)^2) which we used to extend our definition of the ES1 matrix 
               coefficients to m = 0, -1, -2, .. . We discovered that ES1[ -1,n] 
               = 0.5 for n = 1, 2, .. . As usual eta(m) = (1-2^(1-m))*zeta(m) with 
               eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function.
%C A160464 The coefficients in the columns of the ES1 matrix, for m = 1, 2, 3, .. 
               , and n = 2, 3, 4 .. , can be generated with the polynomials GF(z,
               n) for which we found the following general expression GF(z;n) = 
               ((-1)^(n-1)*r(n)*CFN1(z,n)*GF(z;n=1) + ETA(z,n))/p(n).
%C A160464 The CFN1(z,n) polynomials depend on the central factorial numbers A008955.
%C A160464 The ETA(z,n) are the Eta polynomials which lead to the Eta triangle.
%C A160464 The zero patterns of the Eta polynomials resemble a UFO. These patterns 
               resemble those of the Zeta, Beta and Lambda polynomials, see A160474, 
               A160480 and A160487.
%C A160464 The first Maple algorithm generates the coefficients of the Eta triangle. 
               The second Maple algorithm generates the ES1[2*m-1,n] coefficients 
               for m= 0, -1, -2, -3, .. .
%C A160464 The M(n) sequence, see the second Maple algorithm, leads to Gould's sequence 
               A001316 and the 'look-a-like' of the denominators in Taylor series 
               for tan(x), i.e. A156769(n).
%C A160464 Some of our results are conjectures based on numerical evidence, see 
               especially A160466.
%H A160464 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, 
               Chapter 23, pp. 811-812.
%H A160464 J.W. Meijer and N.H.G. Baken, <a href="http://www.sciencedirect.com/science/
               article/B6V1D-47N62TS-4W/2/faab5428aaed95db7cfb6e62c3c93e97">The 
               Exponential Integral Distribution</a>, Statistics and Probability 
               Letters, Volume 5, No.3, April 1987. pp 209-211.
%H A160464 Eric. W. Weisstein, <a href="http://mathworld.wolfram.com/DirichletEtaFunction.html">
               Dirichlet Eta Function</a> from Wolfram MathWorld.
%F A160464 We discovered an interesting relation between the Eta triangle coefficients 
               ETA(n,m) = q(n)*((-1)*ETA(n-1,m-1)+(n-1)^2*ETA(n-1,m)), for n = 3, 
               4, .. and m = 2, 3, .. , with
%F A160464 q(n) = 1+(-1)^(n-3)*(floor(ln(n-1)/ln(2))-floor(ln(n-2)/ln(2))) for n 
               = 3, 4, .. .
%F A160464 See A160465 for ETA(n,m=1) and furthermore ETA(n,n) = 0 for n = 2, 3, 
               .. .
%F A160464 The generating functions GF(z;n) of the coefficients in the matrix columns 
               are defined by
%F A160464 GF(z;n) = sum(ES1[2*m-1,n] * z^(2*m-2), m=1..infinity), with n = 1, 2, 
               3, .. . This leads to
%F A160464 GF(z;n=1) = (2*ln(2)-Psi(z)-Psi(-z)+Psi(1/2*z)+Psi(-1/2*z)); Psi(z) is 
               the digamma-function.
%F A160464 GF(z;n) = ((2*n-2)/(2*n-1)-2*z^2/((n-1)*(2*n-1)))*GF(z;n-1)-1/((n-1)*(2*n-1)).
%F A160464 We found for GF(z;n), for n = 2, 3, .. , the following general expression
%F A160464 GF(z;n) = ((-1)^(n-1)*r(n)*CFN1(z,n)*GF(z;n=1) + ETA(z,n) )/p(n) with
%F A160464 r(n) = 2^floor(log(n-1)/log(2)+1) and
%F A160464 p(n) = 2^(-GCS(n))*(2*n-1)! with
%F A160464 GCS(n) = ln(1/(2^(-(2*(n-1)-1-floor(ln(n-1)/ ln(2))))))/ln(2)
%e A160464 The first few rows of the triangle ETA(n,m) with n=2,3,.. and m=1,2,.. 
               are
%e A160464 [ -1]
%e A160464 [ -11, 2]
%e A160464 [ -114, 29, -2]
%e A160464 [ -3963, 1156, -122, 4]
%e A160464 The first few ETA(z,n) polynomials are
%e A160464 ETA(z,n=2) = -1
%e A160464 ETA(z,n=3) = -11+2*z^2
%e A160464 ETA(z,n=4) = -114+29*z^2-2*z^4
%e A160464 The first few CFN1(z,n) polynomials are
%e A160464 CFN1(z,n=2) = (z^2-1)
%e A160464 CFN1(z,n=3) = (z^4-5*z^2+4)
%e A160464 CFN1(z,n=4) = (z^6- 14*z^4+49*z^2-36)
%e A160464 The first few generating functions GF(z;n) are:
%e A160464 GF(z;n=2) = ((-1)*2*(z^2 - 1)*GF(z;n=1) + (- 1))/3
%e A160464 GF(z;n=3) = (4*(z^4 - 5*z^2+4) *GF(z;n=1) + (-11 + 2*z^2))/30
%e A160464 GF(z;n=4) = ((-1)*4*(z^6-14*z^4+49*z^2-36)*GF(z;n=1) + (-114+29*z^2-2*z^4))/
               315
%p A160464 restart; nmax:=8; mmax:=nmax: c(2):=-1/3: for n from 3 to nmax do c(n):=(2*n-2)*c(n-1)/
               (2*n-1)-1/ ((n-1)*(2*n-1)) end do: for n from 2 to nmax do GCS(n-1):=ln(1/
               (2^(-(2*(n-1)-1-floor(ln(n-1)/ ln(2))))))/ln(2) od: for n from 2 
               to nmax do p(n):=2^(-GCS(n-1))*(2*n-1)! od: for n from 2 to nmax 
               do ETA(n,1):=p(n)*c(n) end do: for n from 2 to mmax do ETA(n,n):=0 
               end do: for m from 2 to mmax do for n from m+1 to nmax do q(n):=(1+(-1)^(n-3)*(floor(ln(n-1)/
               ln(2))- floor(ln(n-2)/ln(2)))): ETA(n,m):= q(n)*((-1)*ETA(n-1,m-1)+(n-1)^2*ETA(n-1,
               m)) end do end do: for n from 2 to nmax do for m from 1 to n-1 do 
               b((((n-2)*(n-1)/2))+m+1) :=ETA(n,m) end do end do: a:=n-> b(n): seq(a(n), 
               n=2..(nmax-1)*(nmax)/2+1);
%p A160464 restart; nmax:=20; m:=1; ES1row:=1-2*m; jn:=nmax: im:=nmax: for n from 
               1 to nmax do for j from 1 to jn do cfn1[1, j]:=1 end do: for i from 
               2 to im do cfn1[i, 1]:=0 end do: for j from 2 to jn do for i from 
               2 to im do cfn1[i, j]:=cfn1[i-1, j-1]*(j-1)^2+cfn1[i, j-1] end do 
               end do: end do: mmax:=nmax: for m1 from 1 to mmax do M(m1-1):=2^(2*m1-2)/
               ((2*m1-1)!) end do: for m1 from 1 to mmax do ES1[ -2*m1+1,1]:=2*(1-2^(1-(1-2*m1)))*(-bernoulli(2*m1)/
               (2*m1)) od: for n from 2 to nmax do for m1 from 1 to mmax-n+1 do 
               ES1[1-2*m1,n]:= (-1)^(n-1)*M(n-1)*sum((-1)^(k+1)*cfn1[k,n]* ES1[2*k 
               -2*n-2*m1+1,1],k=1..n) od: od: a:=n-> ES1[1-2*m,n]: seq(a(n), n=1..nmax-m+1);
%Y A160464 The r(n) sequence equals A062383 (n>=1).
%Y A160464 The p(n) sequence equals A160473(n) (n>=2).
%Y A160464 The GCS(n) sequence equals the Geometric Connell sequence A049039(n).
%Y A160464 The M(n-1) sequence equals A001316(n-1)/A156769(n) (n>=1).
%Y A160464 The q(n) sequence leads to A081729 and the 'gossip sequence' A007456.
%Y A160464 The first right hand column equals A053644 (n>=1).
%Y A160464 The first left hand column equals A160465.
%Y A160464 The row sums equal A160466.
%Y A160464 The CFN1(z, n) and the cfn1(m, n) lead to A008955.
%Y A160464 Cf. A094665 and A160468.
%Y A160464 Cf. The Zeta, Beta and Lambda triangles A160474, A160480 and A160487.
%Y A160464 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 
               2009: (Start)
%Y A160464 Cf. A162440 (EG1 matrix)
%Y A160464 (End)
%Y A160464 Sequence in context: A089365 A130217 A096044 this_sequence A038316 A139311 
               A140749
%Y A160464 Adjacent sequences: A160461 A160462 A160463 this_sequence A160465 A160466 
               A160467
%K A160464 easy,sign,tabl
%O A160464 2,2
%A A160464 Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009

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