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%I #14 Aug 13 2020 09:07:26
%S 5,66,680,2576,33408,14080,545792,481280,29523968,73465856,27525120,
%T 856162304,1153433600,18798870528,86603988992,2080374784,
%U 2385854332928,3216930504704,71829033058304,7593502179328,281749854617600
%N Numerators of the BG1[ -5,n] coefficients of the BG1 matrix
%C 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.
%C We discovered that the n-th 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.
%C 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.
%C 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.
%C The BG1 matrix is related to the BS1 matrix, see A160480 and the formulas below.
%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.
%F 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)!.
%F The generating functions GFB(z;n) of the coefficients in the matrix columns are defined by
%F GFB(z;n) = sum(BG1[2*m-1,n]*z^(2*m-2), m=1..infinity).
%F 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).
%F 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.
%F BG1[2*m-1,n] = (n-1)!^2*4^(n-1)*BS1[2*m-1,n]/(2*n-2)!
%e The first few formulas for the BG1[1-2*m,n] matrix coefficients are:
%e BG1[ -1,n] = (1)*4^(n-1)*(n-1)!^2/(2*n-2)!
%e BG1[ -3,n] = (1-2*n)*4^(n-1)*(n-1)!^2/(2*n-2)!
%e BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!
%e The first few generating functions GFB(z;n) are:
%e GFB(z;2) = ((-1)*(z^2-1)*GFB(z;1) + (-1))/1
%e GFB(z;3) = ((+1)*(z^4-10*z^2+9)*GFB(z;1) + (-11 + z^2))/9
%e GFB(z;4) = ((-1)*( z^6- 35*z^4+259*z^2-225)*GFB(z;1) + (-299 + 36*z^2 - z^4))/225
%p a := proc(n): numer((1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!) end proc: seq(a(n), n=1..21);
%p # End program 1
%p nmax1 := 5; coln := 3; Digits := 20: mmax1 := nmax1: for n from 0 to nmax1 do t2(n, 0) := 1 od: for n from 0 to nmax1 do t2(n, n) := doublefactorial(2*n-1)^2 od: for n from 1 to nmax1 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 mmax1 do BG1[1-2*m, 1] := euler(2*m-2) od: for m from 1 to mmax1 do BG1[2*m-1, 1] := Re(evalf(2*sum((-1)^k1/(1+2*k1)^(2*m), k1=0..infinity))) od: for m from -mmax1 +coln to mmax1 do BG1[2*m-1, coln] := (-1)^(coln+1)*sum((-1)^k1*t2(coln-1, k1)*BG1[2*m-(2*coln-1)+2*k1, 1], k1=0..coln-1)/doublefactorial(2*coln-3)^2 od;
%p # End program 2
%p # Maple programs edited by _Johannes W. Meijer_, Sep 25 2012
%Y A162444 are the denominators of the BG1[ -5, n] matrix coefficients.
%Y The BG1[ -3, n] equal (-1)*A002595(n-1)/A055786(n-1) for n >= 1.
%Y The BG1[ -1, n] equal A046161(n-1)/A001790(n-1) for n >= 1.
%Y The cs(n) equal A046161(n-2)/A001803(n-2) for n >= 2.
%Y The BETA(z, n) polynomials and the BS1 matrix lead to the Beta triangle A160480.
%Y The CFN2(z, n), the t2(n, m) and the BG2 matrix lead to A008956.
%Y Cf. A000364, A001818 and A160485.
%Y Cf. A162443 (BG1 matrix), A162446 (ZG1 matrix) and A162448 (LG1 matrix).
%K easy,frac,nonn
%O 1,1
%A _Johannes W. Meijer_, Jul 06 2009
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