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%I A160474
%S A160474 1,51,10,10594,2961,210,356487,115940,12642,420,101141295,
%T A160474 35804857,4751890,254562,4620,48350824787,18071509911,2689347661,
%U A160474 180909586,5471466,60060
%V A160474 -1,51,-10,-10594,2961,-210,356487,-115940,12642,-420,-101141295,
%W A160474 35804857,-4751890,254562,-4620,48350824787,-18071509911,2689347661,
%X A160474 -180909586,5471466,-60060
%N A160474 The Zeta triangle
%C A160474 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).
%C A160474 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 formulae 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.
%C A160474 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).
%C A160474 The CFN1(z;n) polynomials depend on the central factorial numbers A008955.
%C A160474 The ZETA(z;n) are the Zeta polynomials which lead to the Zeta triangle.
%C A160474 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.
%C A160474 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, .. .
%C A160474 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 A160474 Some of our results are conjectures based on numerical evidence.
%H A160474 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 A160474 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.
%F A160474 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, .. .
%F A160474 We observe that the ZL(n) = A160479(n) sequence also rules the Lambda 
               triangle A160487.
%F A160474 The generating functions GH(z;n) of the coefficients in the matrix columns 
               are defined by
%F A160474 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, .. .
%F A160474 We found the following general expression for the GH(z;n) polynomials, 
               for n = 2, 3, ..
%F A160474 GH(z;n) = (h(n)*CFN1(z;n)*GH(z;n=1) + ZETA(z;n))/p(n) with
%F A160474 h(n) = 6*A160476(n) and p(n) = A160478(n).
%e A160474 The first few rows of the triangle ZETA(n,m) with n=2,3,.. and m=1,2,
               .. are
%e A160474 [ -1]
%e A160474 [51, -10]
%e A160474 [ -10594, 2961, -210]
%e A160474 [356487, -115940, 12642, -420]
%e A160474 The first few ZETA(z;n) polynomials are
%e A160474 ZETA(z;n=2) = -1
%e A160474 ZETA(z;n=3) = 51 -10*z^2
%e A160474 ZETA(z;n=4) = -10594+2961*z^2-210*z^4
%e A160474 The first few CFN1(z;n) polynomials are
%e A160474 CFN1(z;n=2) = (z^2-1)
%e A160474 CFN1(z;n=3) = (z^4-5*z^2+4)
%e A160474 CFN1(z;n=4) = (z^6- 14*z^4+49*z^2-36)
%e A160474 The first few generating functions GH(z;n) are:
%e A160474 GH(z;n=2) = (6*(z^2-1)*GH(z;n=1) + (-1)) / 9
%e A160474 GH(z;n=3) = (60*(z^4-5*z^2+4)*GH(z;n=1) + (51 -10*z^2)) / 450
%e A160474 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
%p A160474 restart; nmax:=7: jn:=nmax: im:=nmax: Omega(0):=1: 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: 
               Omega(n):= (sum((-1)^(k+n+1)*(bernoulli(2*k)/(2*k))*cfn1(n-k+1, n), 
               k=1..n))/(2*n-1)! end do: for n from 1 to nmax do d(n):=(Omega(n)*2^(2*n-1)) 
               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:
%p A160474 y(1):=Zc(1): for n from 1 to nmax-2 do y(n+1):= Zc(n+1)-((2*n+2)/(2*n+3))*y(n) 
               end do: for n from 2 to nmax do ZETA(n,1):= p(n)*y(n-1) end do: nmax:=nmax; 
               mmax:=nmax: for n from 2 to nmax do ZETA(n,n):=0 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:
%p A160474 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 d(n+2):=cm(n+1)/cm(n) end do: for m from 2 
               to mmax do for n from m+1 to nmax do ZETA(n,m):= d(n)*(ZETA(n-1,m-1)-(n-1)^2* 
               ZETA(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):=ZETA(n,m) end do end do: a:=n-> 
               b(n): seq(a(n), n=2..(nmax-1)*(nmax)/2+1);
%p A160474 restart; nmax:=10; m:=1; ZS1row:=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 ZS1[ -2*m1+1,1]:=2*(-bernoulli(2*m1)/
               (2*m1)) od: for n from 2 to nmax do for m1 from 1 to mmax-n+1 do 
               ZS1[ -2*m1+1,n]:= M(n-1)*sum((-1)^(k+1)* cfn1[k,n]* ZS1[2*k -2*n-2*m1 
               +1,1],k=1..n) od: od: a:=n-> ZS1[1-2*m,n]: seq(a(n), n=1..nmax-m+1);
%Y A160474 A160475 equals the first left hand column.
%Y A160474 A160476 equals the first right hand column and 6*h(n).
%Y A160474 A160477 equals the rows sums .
%Y A160474 A160478 equals the p(n) sequence.
%Y A160474 A160479 equals the ZL(n) sequence.
%Y A160474 A001620 is the Euler-Mascheroni constant gamma.
%Y A160474 The M(n-1) sequence equals A001316(n-1)/A156769(n) (n>=1).
%Y A160474 The ZS1[ -1, n] coefficients lead to A002195 and A002196.
%Y A160474 The CFN1(z, n) and the cfn1(m, n) lead to A008955.
%Y A160474 Cf. The Eta, Beta and Lambda triangles A160464, A160480 and A160487.
%Y A160474 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 
               2009: (Start)
%Y A160474 Cf. A162446 (ZG1 matrix)
%Y A160474 (End)
%Y A160474 Sequence in context: A152515 A111402 A087408 this_sequence A033371 A131536 
               A003912
%Y A160474 Adjacent sequences: A160471 A160472 A160473 this_sequence A160475 A160476 
               A160477
%K A160474 easy,sign,tabl
%O A160474 2,2
%A A160474 Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009

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