cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A160487 The Lambda triangle.

Original entry on oeis.org

1, -107, 10, 59845, -7497, 210, -6059823, 854396, -35574, 420, 5508149745, -827924889, 41094790, -765534, 4620, -8781562891079, 1373931797082, -75405128227, 1738417252, -17219202, 60060
Offset: 2

Views

Author

Johannes W. Meijer, May 24 2009, Sep 18 2012

Keywords

Comments

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 formulas enable us to determine the values of the LS1[2*m,n] coefficients, for all integers m and all positive integers n, but not for all n. 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.

Examples

			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

References

  • 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.

Links

  • 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.
  • Johannes W. Meijer, The zeros of the Eta, Zeta, Beta and Lambda polynomials, jpg and pdf, Mar 03 2013.

Crossrefs

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(n, k) lead to A008956.
Cf. The Eta, Zeta and Beta triangles A160464, A160474 and A160480.
Cf. A162448 (LG1 matrix)

Programs

  • Maple
    nmax:=7; for n from 0 to nmax do cfn2(n, 0) := 1: cfn2(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax do for k from 1 to n-1 do cfn2(n, k) := (2*n-1)^2*cfn2(n-1, k-1) + cfn2(n-1, k) od: od: for n from 1 to nmax do Delta(n-1) := sum((1-2^(2*k1-1))* (-1)^(n+1)*(-bernoulli(2*k1)/(2*k1))*(-1)^(k1+n)*cfn2(n-1,n-k1, n), k1=1..n) / (2*4^(n-1)*(2*n-1)!); LAMBDA(-2, n) := sum(2*(1-2^(2*k1-1))*(-bernoulli(2*k1) / (2*k1))*(-1)^(k1+n)* cfn2(n-1,n-k1), k1=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; seq(seq(LAMBDA(n,m), m=1..n-1), n=2..nmax);
# End first program.
nmax1:=10; m:=1; LS1row:=-2*m; for n from 0 to nmax1 do cfn2(n, 0) := 1: cfn2(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax1 do for k from 1 to n-1 do cfn2(n, k) := (2*n-1)^2*cfn2(n-1, k-1) + cfn2(n-1, k) od: od: mmax1:=nmax1: for m1 from 1 to mmax1 do LS1[-2*m1, 1] := 2*(1-2^(-(-2*m1+1)))*(-bernoulli(2*m1)/(2*m1)) od: for n from 2 to nmax1 do for m1 from 1 to mmax1-n+1 do LS1[ -2*m1, n] := sum((-1)^(k1+1)*cfn2(n-1,k1-1)* LS1[2*k1-2*n-2*m1, 1], k1=1..n)/(2*n-2)! od: od: seq(LS1[ -2*m, n], n=1..nmax1-m+1);
# End second program.

Formula

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).

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.