1,1

The BG1 matrix coefficients are defined by BG1[2m-1,1] = 2*beta(2m) and the recurrence relation BG1[2m-1,n] = BG1[2m-1,n-1] - BG1[2m-3,n-1]/(2*n-3)^2 with m = .. , -2, -1, 0, 1, 2, .. and n = 1, 2, 3, .. . As usual beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). For the BG2 matrix, the even counterpart of the BG1 matrix, see A008956.

We discovered that the nth term of the row coefficients can be generated with BG1[1-2*m,n] = RBS1(1-2*m,n)* 4^(n-1)*((n-1)!)^2/ (2*n-2)! for m => 1. For the BS1(1-2*m,n) polynomials see A160485.

The coefficients in the columns of the BG1 matrix, for m =>1 and n =>2, can be generated with GFB(z;n) = ((-1)^(n+1)*CFN2(z;n)*GFB(z;n=1) + BETA(z;n))/((2*n-3)!!)^2 for n =>2. For the CFN2(z;n) and the Beta polynomials see A160480.

The BG1[ -5,n] sequence can be generated with the first Maple program and the BG1[2*m-1,n] matrix coefficients can be generated with the second Maple program.

The BG1 matrix is related to the BS1 matrix, see A160480 and the formulae below.

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

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.

a(n) = numer(BG1[ -5,n]) and A162444(n) = denom(BG1[ -5,n]) with BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!.

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

GFB(z;n) = sum(BG1[2*m-1,n]*z^(2*m-2), m=1..infinity).

GFB(z;n) = (1-z^2/(2*n-3)^2)*GFB(n-1) - 4^(n-2)*(n-2)!^2/((2*n-4)!*(2*n-3)^2) for n => 2 with GFB(z;n=1) = 1/(z*cos(Pi*z/2))*int(sin(z*t)/sin(t),t=0..Pi/2).

The column sums cs(n) = sum(BG1[2*m-1,n]*z^(2*m-2), m=1..infinity) = 4^(n-1)/((2*n-2)*binomial(2*n-2,n-1)) for n => 2.

BG1[2*m-1,n] = (n-1)!^2*4^(n-1)*BS1[2*m-1,n]/(2*n-2)!

The first few formulae for the BG1[1-2*m,n] matrix coefficients are:

BG1[ -1,n] = (1)*4^(n-1)*(n-1)!^2/(2*n-2)!

BG1[ -3,n] = (1-2*n)*4^(n-1)*(n-1)!^2/(2*n-2)!

BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!

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

GFB(z;2) = ((-1)*(z^2-1)*GFB(z;1) + (-1))/1

GFB(z;3) = ((+1)*(z^4-10*z^2+9)*GFB(z;1) + (-11 + z^2))/9

GFB(z;4) = ((-1)*( z^6- 35*z^4+259*z^2-225)*GFB(z;1) + (-299 + 36*z^2 - z^4))/225

restart; nmax:=23; seq(numer((1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!), n=1..nmax);

restart; nmax:=5; coln:=3; Digits:=20: mmax:=nmax: for n from 0 to nmax do t2(n, 0):=1 od: for n from 0 to nmax do t2(n, n) := doublefactorial(2*n-1)^2 od: for n from 1 to nmax do for m from 1 to n-1 do t2(n, m):=(2*n-1)^2* t2(n-1, m-1)+ t2(n-1, m) od: od: for m from 1 to mmax do BG1[1-2*m, 1] := euler(2*m-2) od: for m from 1 to mmax do BG1[2*m-1, 1]:= Re(evalf(2*sum((-1)^k/(1+2*k)^(2*m), k=0..infinity))) od: for m from -mmax +coln to mmax do BG1[2*m-1, coln] := (-1)^(coln+1)*sum((-1)^k*t2(coln-1, k)*BG1[2*m-(2*coln-1)+2*k, 1], k=0..coln-1)/doublefactorial(2*coln-3)^2 od;

A162444 are the denominators of the BG1[ -5, n] matrix coefficients.

The BG1[ -3, n] equal (-1)*A002595(n-1)/A055786(n-1) for n=>1.

The BG1[ -1, n] equal A046161(n-1)/A001790(n-1) for n=>1.

The cs(n) equal A046161(n-2)/A001803(n-2) for n=>2.

The BETA(z, n) polynomials and the BS1 matrix lead to the Beta triangle A160480.

The CFN2(z, n), the t2(n, m) and the BG2 matrix lead to A008956.

Cf. A000364, A001818 and A160485.

Cf. A162443 (BG1 matrix), A162446 (ZG1 matrix) and A162448 (LG1 matrix).

Sequence in context: A012476 A167536 A042491 this_sequence A156597 A059489 A059852

Adjacent sequences: A162440 A162441 A162442 this_sequence A162444 A162445 A162446

easy,frac,nonn

Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 2009