%I #31 May 05 2023 07:47:48

%S -1,-11,1,-299,36,-1,-15371,2063,-85,1,-1285371,182474,-8948,166,-1,

%T -159158691,23364725,-1265182,29034,-287,1,-27376820379,4107797216,

%U -237180483,6171928,-77537,456,-1

%N The Beta triangle read by rows.

%C 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, ... .

%C 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).

%C 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).

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

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

%C 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.

%C 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, ... .

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

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.

%H J. M. Amigo, <a href="https://doi.org/10.1155/2008/421478"> Relations among Sums of Reciprocal Powers Part II</a>, International Journal of Mathematics and Mathematical Sciences , Volume 2008 (2008), pp. 1-20.

%H Johannes W. Meijer, The zeros of the Eta, Zeta, Beta and Lambda polynomials, <a href="/A160480/a160480.jpg">jpg</a> and <a href="/A160480/a160480.pdf">pdf</a>, Mar 03 2013.

%F 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, ... .

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

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

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

%F 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, ... .

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

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

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

%e [ -1],

%e [ -11, 1],

%e [ -299, 36, -1],

%e [ -15371, 2063 -85, 1].

%e The first few BETA(z;n) polynomials are

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

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

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

%e The first few CFN1(z;n) polynomials are

%e CFN2(z;n=2) = (z^2 - 1),

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

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

%e The first few generating functions GK(z;n) are

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

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

%e 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.

%p 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: seq(seq(BETA(n, m), m=1..n-1), n= 2..nmax);

%p # End first program

%p nmax1 := 25; m := 1; BS1row := 1-2*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 BS1[1-2*m1, 1] := euler(2*m1-2) od: for n from 2 to nmax1 do for m1 from 1 to mmax1-n+1 do BS1[1-2*m1, n] := (-1)^(n+1)*sum((-1)^(k1+1)*cfn2(n-1, k1-1) * BS1[2*k1-2*n-2*m1+1, 1], k1 =1..n)/(2*n-2)! od: od: seq(BS1[1-2*m, n], n=1..nmax1-m+1);

%p # End second program

%t BETA[2, 1] = -1;

%t BETA[n_, 1] := BETA[n, 1] = (2*n - 3)^2*BETA[n - 1, 1] - (2*n - 4)!;

%t 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];

%t BETA[_, _] = 0;

%t Table[BETA[n, m], {n, 2, 9}, {m, 1, n - 1}] // Flatten (* _Jean-François Alcover_, Dec 13 2017 *)

%Y A160481 equals the rows sums.

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

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

%Y A160485 and A160486 are two related triangles.

%Y The CFN2(z, n) and the cfn2(n, k) lead to A008956.

%Y Cf. the Eta, Zeta and Lambda triangles: A160464, A160474 and A160487.

%Y Cf. A162443 (BG1 matrix).

%K uned,easy,sign,tabl

%O 2,2

%A _Johannes W. Meijer_, May 24 2009, Sep 19 2012