proposed

approved

editing

proposed

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 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, .. . .

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, .. . .

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

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].

BETA(z;n=2) = -1,

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

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

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.

A160481 equals the rows sums .

Cf. A162443 (BG1 matrix).

approved

editing

reviewed

approved

proposed

reviewed

editing

proposed

BETA[2, 1] = -1;

BETA[2, 1] = -1; BETA[n_, 1] := BETA[n, 1] = (2*n - 3)^2*BETA[n - 1, 1] - (2*n - 4)!; BETA[n_ /; n > 2, m_ /; m > 0] /; 1 <= m <= n := BETA[n, m] = (2*n - 3)^2*BETA[n - 1, m] - BETA[n - 1, m - 1]; BETA[_, _] = 0;

BETA[n_ /; n > 2, m_ /; m > 0] /; 1 <= m <= n := BETA[n, m] = (2*n - 3)^2*BETA[n - 1, m] - BETA[n - 1, m - 1];

BETA[_, _] = 0;

uned,easy,sign,tabl

approved

editing

reviewed

approved

proposed

reviewed

editing

proposed