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%I A160487
%S A160487 1,107,10,59845,7497,210,6059823,854396,35574,420,5508149745,
%T A160487 827924889,41094790,765534,4620,8781562891079,1373931797082,
%U A160487 75405128227,1738417252,17219202,60060
%V A160487 1,-107,10,59845,-7497,210,-6059823,854396,-35574,420,5508149745,
%W A160487 -827924889,41094790,-765534,4620,-8781562891079,1373931797082,
%X A160487 -75405128227,1738417252,-17219202,60060
%N A160487 The Lambda triangle
%C A160487 The coefficients of the LS1 matrix are defined by LS1[2*m,n] = int(y^(2*m)/
               (sinh(y))^(2*n-1),y=0..infinity)/factorial(2*m) for m = 1, 2, 3, 
               .. and n = 1, 2, 3, .. under the condition that n <= m.
%C A160487 This definition leads to LS1[2*m,n=1] = 2*lambda(2*m+1), for m = 1, 2, 
               .. , and the recurrence relation LS1[2*m,n] = ((2*n-3)/(2*n-2))*(LS1[2*m-2,
               n-1]/(2*n-3)^2- LS1[2*m,n-1]). As usual lambda(m) = (1-2^(-m))*zeta(m) 
               with zeta(m) the Riemann zeta function.
%C A160487 These two formulae enable us to determine the values of the LS1[2*m,n] 
               coefficients, for m all integers and n all positive integers, but 
               not for all. If we choose, somewhat but not entirely arbitrarily, 
               LS1[m=0,n=1] = gamma, with gamma the Euler-Mascheroni constant, we 
               can determine them all.
%C A160487 The coefficients in the columns of the LS1 matrix, for m = 0, 1, 2, .. 
               , and n = 2, 3, 4 .. , can be generated with the GL(z;n) polynomials 
               for which we found the following general expression GL(z;n) = (h(n)*CFN2(z;
               n)*GL(z;n=1) + LAMBDA(z;n))/p(n).
%C A160487 The CFN2(z;n) polynomials depend on the central factorial numbers A008956.
%C A160487 The LAMBDA(z;n) are the Lambda polynomials which lead to the Lambda triangle.
%C A160487 The zero patterns of the Lambda polynomials resemble a UFO. These patterns 
               resemble those of the Eta, Zeta and Beta polynomials, see A160464, 
               A160474 and A160480.
%C A160487 The first Maple algorithm generates the coefficients of the Lambda triangle. 
               The second Maple algorithm generates the LS1[2*m,n] coefficients 
               for m= -1, -2, -3, .. .
%C A160487 Some of our results are conjectures based on numerical evidence.
%H A160487 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.
%F A160487 We discovered a remarkable relation between the Lambda triangle coefficients 
               Lambda(n,m) = ZL(n)*(Lambda(n-1,m-1)-(2*n-3)^2*Lambda(n-1,m)) for 
               n = 3, 4, .. and m = 2, 3, .. . See A160488 for LAMBDA(n,m=1) and 
               furthermore LAMBDA(n,n) = 0 for n = 2, 3, .. .
%F A160487 We observe that the ZL(n) = A160479(n) sequence also rules the Zeta triangle 
               A160474.
%F A160487 The generating functions GL(z;n) of the coefficients in the matrix columns 
               are defined by
%F A160487 GL(z;n) = sum(LS1[2*m-2,n]*z^(2*m-2), m=1..infinity), with n = 1, 2, 
               3, .. .
%F A160487 This definition, and our choice of LS1[m=0,n=1] = gamma, leads to GL(z;
               n=1) = -2*Psi(1-z)+Psi(1-(z/2))-(Pi/2)*tan(Pi*z/2) with Psi(z) the 
               digamma-function. Furthermore we discovered that GL(z;n) =GL(z;n-1)*(z^2/
               ((2*n-2)*(2*n-3)) -(2*n-3)/((2*n-2)))+LS1[ -2,n-1]/((2*n-2)*(2*n-3)) 
               for n = 2, 3 , .. . with LS1[ -2,n] = (-1)^(n-1)*4*A058962(n-1)*A002197(n-1)/
               A002198(n-1) for n = 1, 2, .. , with A058962(n-1) = 2^(2*n-2)*(2*n-1).
%F A160487 We found the following general expression for the GL(z;n) polynomials, 
               for n = 2, 3, ..
%F A160487 GL(z;n) = (h(n)*CFN2(z;n)*GL(z;n=1) + LAMBDA(z;n))/p(n) with
%F A160487 h(n) = 6*A160476(n) and p(n) = A160490(n).
%e A160487 The first few rows of the triangle LAMBDA(n,m) with n=2,3,.. and m=1,
               2,.. are
%e A160487 [1]
%e A160487 [ -107, 10]
%e A160487 [59845, -7497, 210]
%e A160487 [ -6059823, 854396, -35574, 420]
%e A160487 The first few LAMBDA(z;n) polynomials are
%e A160487 LAMBDA (z;n=2) = 1
%e A160487 LAMBDA (z;n=3) = -107 +10*z^2
%e A160487 LAMBDA (z;n=4) = 59845-7497*z^2+210*z^4
%e A160487 The first few CFN2(z;n) polynomials are
%e A160487 CFN2(z;n=2) = (z^2-1)
%e A160487 CFN2(z;n=3) = (z^4-10*z^2+9)
%e A160487 CFN2(z;n=4) = (z^6- 35*z^4+259*z^2-225)
%e A160487 The first few generating functions GL(z;n) are:
%e A160487 GL(z;n=2) = (6*(z^2-1)*GL(z,n=1) + (1)) /12
%e A160487 GL(z;n=3) = (60*(z^4-10*z^2+9)*GL(z,n=1)+ (-107+10*z^2)) / 1440
%e A160487 GL(z;n=4) = (1260*( z^6- 35*z^4+259*z^2-225)*GL(z,n=1) + (59845-7497*z^2+ 
               210*z^4))/907200
%p A160487 restart; nmax:=7; jn:=nmax+1: im:=nmax+1: for n from 1 to nmax do for 
               i from 2 to im do cfn2(i, 1):=0 end do: for j from 1 to jn do cfn2(1,
               j):=1 end do: for j from 2 to jn do for i from 2 to im do cfn2(i,
               j):= cfn2(i,j-1) + cfn2(i-1,j-1)*(2*j-3)^2 end do end do: Delta(n-1):= 
               sum((1-2^(2*k-1))* (-1)^(n+1)*(-bernoulli(2*k)/(2*k))*(-1)^(k+n)*cfn2(n-k+1,
               n), k=1..n) /(2*4^(n-1)*(2*n-1)!); LAMBDA(-2,n):= sum(2*(1-2^(2*k-1))*(-bernoulli(2*k)/ 
               (2*k))*(-1)^(k+n)* cfn2(n+1-k,n),k=1..n)/ factorial(2*n-2) end do: 
               Lcgz(2):=1/12: f(2):=1/12: for n from 3 to nmax do Lcgz(n):=LAMBDA(-2,
               n-1)/((2*n-2)*(2*n-3)):
%p A160487 f(n):= Lcgz(n)-((2*n-3)/(2*n-2))*f(n-1) end do: for n from 1 to nmax 
               do b(n):=denom(Lcgz(n+1)) end do: for n from 1 to nmax do b(n):=2*n*denom(Delta(n-1))/
               2^(2*n) end do: p(2):=b(1): for n from 2 to nmax do p(n+1):= lcm(p(n)*(2*n)*(2*n-1),
               b(n)) end do: for n from 2 to nmax do LAMBDA(n,1) :=p(n)*f(n) end 
               do: mmax:=nmax: for n from 2 to nmax do LAMBDA(n,n):=0 end do: for 
               n from 1 to nmax 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 nmax-1 do c(n+1):=lcm(c(n)*(2*n+2)* 
               (2*n+1),b(n+1)) end do: for n from 1 to nmax do cm(n):= c(n)/(6*(2*n)!) 
               end do:
%p A160487 for n from 1 to nmax-1 do ZL(n+2):=cm(n+1)/cm(n) end do: for m from 2 
               to mmax do for n from m+1 to nmax do LAMBDA(n,m):= ZL(n)*(LAMBDA(n-1,
               m-1)-(2*n-3)^2*LAMBDA(n-1,m)) end do end do; for m from 1 to mmax-1 
               do LAMBDA(n, m) 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):= LAMBDA(n,m) end do end do: a:=n-> 
               b(n): seq(a(n), n=2..(nmax-1)*(nmax)/2+1);
%p A160487 restart; nmax:=10; m:=1; LS1row:=-2*m; jn:=nmax: im:=nmax: for n from 
               1 to nmax do for i from 2 to im do cfn2[i,1]:=0 end do: for j from 
               1 to jn do cfn2[1,j]:=1 end do: for j from 2 to jn do for i from 
               2 to im do cfn2[i,j]:= cfn2[i,j-1] + cfn2[i-1,j-1]*(2*j-3)^2 end 
               do end do: end do: mmax:=nmax: for m1 from 1 to mmax do LS1[ -2*m1,
               1]:=2*(1-2^(-(-2*m1+1)))*(-bernoulli(2*m1)/(2*m1)) od: for n from 
               2 to nmax do for m1 from 1 to mmax-n+1 do LS1[ -2*m1,n]:= sum((-1)^(k+1)*cfn2[k,
               n]* LS1[2*k-2*n-2*m1,1],k=1..n)/(2*n-2)! od: od: a:=n-> LS1[ -2*m,
               n]: seq(a(n), n=1..nmax-m+1);
%Y A160487 A160488 equals the first left hand column.
%Y A160487 A160476 equals the first right hand column and 6*h(n).
%Y A160487 A160489 equals the rows sums .
%Y A160487 A160490 equals the p(n) sequence.
%Y A160487 A160479 equals the ZL(n) sequence.
%Y A160487 A001620 is the Euler-Mascheroni constant gamma.
%Y A160487 The LS1[ -2, n] coefficients lead to A002197, A002198 and A058962.
%Y A160487 The LS1[ -2*m, 1] coefficients equal (-1)^(m+1)*A036282/A036283.
%Y A160487 The CFN2(z, n) and the cfn2(m, n) lead to A008956.
%Y A160487 Cf. The Eta, Zeta and Beta triangles A160464, A160474 and A160480.
%Y A160487 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 
               2009: (Start)
%Y A160487 Cf. A162448 (LG1 matrix)
%Y A160487 (End)
%Y A160487 Sequence in context: A078281 A082177 A156020 this_sequence A096712 A161176 
               A161372
%Y A160487 Adjacent sequences: A160484 A160485 A160486 this_sequence A160488 A160489 
               A160490
%K A160487 easy,sign,tabl
%O A160487 2,2
%A A160487 Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009

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