2,2

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.

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.

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.

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).

The CFN2(z;n) polynomials depend on the central factorial numbers A008956.

The LAMBDA(z;n) are the Lambda polynomials which lead to the Lambda triangle.

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.

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, .. .

Some of our results are conjectures based on numerical evidence.

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.

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, .. .

We observe that the ZL(n) = A160479(n) sequence also rules the Zeta triangle A160474.

The generating functions GL(z;n) of the coefficients in the matrix columns are defined by

GL(z;n) = sum(LS1[2*m-2,n]*z^(2*m-2), m=1..infinity), with n = 1, 2, 3, .. .

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).

We found the following general expression for the GL(z;n) polynomials, for n = 2, 3, ..

GL(z;n) = (h(n)*CFN2(z;n)*GL(z;n=1) + LAMBDA(z;n))/p(n) with

h(n) = 6*A160476(n) and p(n) = A160490(n).

The first few rows of the triangle LAMBDA(n,m) with n=2,3,.. and m=1,2,.. are

[1]

[ -107, 10]

[59845, -7497, 210]

[ -6059823, 854396, -35574, 420]

The first few LAMBDA(z;n) polynomials are

LAMBDA (z;n=2) = 1

LAMBDA (z;n=3) = -107 +10*z^2

LAMBDA (z;n=4) = 59845-7497*z^2+210*z^4

The first few CFN2(z;n) polynomials are

CFN2(z;n=2) = (z^2-1)

CFN2(z;n=3) = (z^4-10*z^2+9)

CFN2(z;n=4) = (z^6- 35*z^4+259*z^2-225)

The first few generating functions GL(z;n) are:

GL(z;n=2) = (6*(z^2-1)*GL(z,n=1) + (1)) /12

GL(z;n=3) = (60*(z^4-10*z^2+9)*GL(z,n=1)+ (-107+10*z^2)) / 1440

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

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)):

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:

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);

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);

A160488 equals the first left hand column.

A160476 equals the first right hand column and 6*h(n).

A160489 equals the rows sums .

A160490 equals the p(n) sequence.

A160479 equals the ZL(n) sequence.

A001620 is the Euler-Mascheroni constant gamma.

The LS1[ -2, n] coefficients lead to A002197, A002198 and A058962.

The LS1[ -2*m, 1] coefficients equal (-1)^(m+1)*A036282/A036283.

The CFN2(z, n) and the cfn2(m, n) lead to A008956.

Cf. The Eta, Zeta and Beta triangles A160464, A160474 and A160480.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 2009: (Start)

Cf. A162448 (LG1 matrix)

(End)

Sequence in context: A078281 A082177 A156020 this_sequence A096712 A161176 A161372

Adjacent sequences: A160484 A160485 A160486 this_sequence A160488 A160489 A160490

easy,sign,tabl

Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009