2,2

The coefficients of the BS1 matrix are defined by BS1[2*m-1,n] = int(y^(2*m-1)/(cosh(y))^(2*n-1),y=0..infinity)/factorial(2*m-1) for m = 1, 2, .. and n = 1, 2, .. .

This definition leads to BS1[2*m-1,n=1] = 2*beta(2*m), for m = 1, 2, .. , and the recurrence relation BS1 [2*m-1,n] = (2*n-3)/(2*n-2)*(BS1[2*m-1,n-1]- BS1[2*m-3,n-1]/(2*n-3)^2) which we used to extend our definition of the BS1 matrix coefficients to m = 0, -1, -2, .. . We discovered that BS1[ -1,n] = 1 for n = 1, 2, .. . As usual beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity).

The coefficients in the columns of the BS1 matrix, for m = 1, 2, 3, .. , and n = 2, 3, 4 .. , can be generated with the GK(z;n) polynomials for which we found the following general expression GK(z;n) = ((-1)^(n+1)*CFN2(z;n)*GK(z;n=1) + BETA(z;n))/p(n).

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

The BETA(z;n) are the Beta polynomials which lead to the Beta triangle.

The zero patterns of the Beta polynomials resemble a UFO. These patterns resemble those of the Eta, Zeta and Lambda polynomials, see A160464, A160474 and A160487.

The first Maple algorithm generates the coefficients of the Beta triangle. The second Maple algorithm generates the BS1[2*m-1,n] coefficients for m= 0, -1, -2, -3, .. .

Some of our results are conjectures based on numerical evidence, see especially A160481.

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.

J.M. Amigo, Relations among Sums of Reciprocal Powers Part II, International Journal of Mathematics and Mathematical Sciences , Volume 2008 (2008), pp. 1-20.

We discovered a relation between the Beta triangle coefficients BETA(n,m) = (2*n-3)^2* BETA(n-1,m)- BETA(n-1,m-1) for n = 3, 4, .. and m = 2, 3, .. with BETA(n,m=1) = (2*n-3)^2* BETA(n-1,m=1)-(2*n-4)! for n = 2, 3, .. and BETA(n,n) = 0 for n = 1, 2, .. .

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

GK(z;n) = sum(BS1[2*m-1,n]*z^(2*m-2), m=1..infinity) with n = 1, 2, .. .

This definition leads to GK(z;n=1) = 1/(z*cos(Pi*z/2))*int(sin(z*t)/sin(t),t=0..Pi/2).

Furthermore we discovered that GK(z;n) = GK(z;n-1)*((2*n-3)/(2*n-2)-z^2/((2*n-2)*(2*n-3)))-1/((2*n-2)*(2*n-3)) for n = 2, 3 , .. .

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

GK(z;n) = ((-1)^(n+1)*CFN2(z;n)*GK(z;n=1) + BETA(z;n))/p(n) with p(n) = (2*n-2)!.

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

[ -1]

[ -11, 1]

[ -299, 36, -1]

[ -15371, 2063 -85, 1]

The first few BETA(z;n) polynomials are

BETA(z;n=2) = -1

BETA(z;n=3) = -11 + z^2

BETA(z;n=4) = -299 + 36*z^2 - z^4

The first few CFN1(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 GK(z;n) are:

GK(z;n=2) = ((-1)*(z^2-1)*GK(z,n=1) + (-1))/2

GK(z;n=3) = ((z^4-10*z^2+9)*GK(z,n=1)+ (-11 + z^2))/24

GK(z;n=4) = ((-1)*( z^6- 35*z^4+259*z^2-225)*GK(z,n=1) + (-299 + 36*z^2 - z^4))/720

restart; nmax:=8; mmax:=nmax: for n from 1 to nmax do BETA(n, n):=0 end do: m:=1: for n from m+1 to nmax do BETA(n, m):=(2*n-3)^2*BETA(n-1, m)-(2*n-4)! od: for m from 2 to mmax do for n from m+1 to nmax do BETA(n, m):=(2*n-3)^2*BETA(n-1, m)-BETA(n-1, m-1) od: od: for n from 2 to nmax do for m from 1 to n-1 do b((((n-2)*(n-1)/2))+m+1):=BETA(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; BS1row:=1-2*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 BS1[1-2*m1, 1]:= euler(2*m1-2) od: for n from 2 to nmax do for m1 from 1 to mmax-n+1 do BS1[1-2*m1, n]:= (-1)^(n+1)*sum((-1)^(k+1)*cfn2[k, n]* BS1[2*k-2*n-2*m1+1, 1], k=1..n)/(2*n-2)! od: od: a:=n-> BS1[1-2*m, n]: seq(a(n), n=1..nmax-m+1);

A160481 equals the rows sums .

A101269 and A160482 equal the first and second left hand columns.

A160483 and A160484 equal the second and third right hand columns.

A160485 and A160486 are two related triangles.

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

Cf. The Eta, Zeta and Lambda triangles A160464, A160474 and A160487.

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

Cf. A162443 (BG1 matrix).

(End)

Sequence in context: A038315 A093158 A132098 this_sequence A045998 A027645 A010190

Adjacent sequences: A160477 A160478 A160479 this_sequence A160481 A160482 A160483

easy,sign,tabl

Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009, Jun 29 2009