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Numerators of the column sums of the LG1 matrix
8

%I #8 May 08 2018 15:11:56

%S -11,863,-215641,41208059,-9038561117,28141689013943,

%T -2360298440602051,3420015713873670001,-147239749512798268300237,

%U 176556159649301309969405807,-178564975300377173768513546347

%N Numerators of the column sums of the LG1 matrix

%C The LG1 matrix coefficients are defined by LG1[2m,1] = 2*lambda(2m+1) for m = 1, 2, .. , and the recurrence relation LG1[2*m,n] = LG1[2*m-2,n-1]/((2*n-3)*(2*n-1)) - (2*n-3)*LG1[2*m,n-1]/(2*n-1) with m = .. , -2, -1, 0, 1, 2, .. and n = 1, 2, 3, .. , under the condition that n <= m. As usual lambda(m) = (1-2^(-m))*zeta(m) with zeta(m) the Riemann zeta function. For the LG2 matrix, the even counterpart of the LG1 matrix, see A008956.

%C These two formulas enable us to determine the values of the LG1[2*m,n] coefficients, with m all integers and n all positive integers, but not for all. If we choose, somewhat but not entirely arbitrarily, LG1[0,1] = gamma, with gamma the Euler-Mascheroni constant, we can determine them all.

%C The coefficients in the columns of the LG1 matrix, for m >= 1 and n >= 2, can be generated with GFL(z;n) = (hg(n)*CFN2(z;n)*GFL(z;n=1) + LAMBDA(z;n))/pg(n) with pg(n) = 6*(2*n-3)!!*(2*n-1)!!*A160476(n) and hg(n) = 6*A160476(n). For the CFN2(z;n) and the LAMBDA(z;n) see A160487.

%C The values of the column sums cs(n) = sum(LG1[2*m,n], m = 0.. infinity), for n >= 2, can be determined with the first Maple program. In this program we have made use of the remarkable fact that if we take LGx[2*m,n] = 2, for m >= 0, and LGx[ -2,n] = LG1[ -2,n] and assume that the recurrence relation remains the same we find that the column sums of this new matrix converge to the same values as the original cs(n).

%C The LG1[2*m,n] matrix coefficients can be generated with the second Maple program.

%C The LG1 matrix is related to the LS1 matrix, see A160487 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.

%F a(n) = numer(cs(n)) and denom(cs(n)) = A162449(n).

%F with cs(n) = sum(LG1[2*m,n], m = 0 .. infinity) for n >= 2.

%F GFL(z;n) = sum( LG1[2*m,n]*z^(2*m-2),m=1..infinity)

%F GFL(z;n) = (LG1[ -2,n-1])/((2*n-3)*(2*n-1))+(z^2/((2*n-3)*(2*n-1))-(2*n-3)/(2*n-1))*GFL(z;n-1) with GFL(z;n=1) = -2*Psi(1-z)+Psi(1-(z/2))-(Pi/2)*tan(Pi*z/2)

%F LG1[ -2,n] = (-1)^(n+1)*4*(A061549(n-1)/A001790(n-1))*(A002197(n-1)/A002198(n-1))

%F LG1[2*m,n] = (4^(n-1)/((2*n-1)*binomial(2*n-2,n-1)))*LS1[2*m,n]

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

%e GFL(z;2) = (6*(z^2-1)*GFL(z;1)+(1))/18

%e GFL(z;3) = (60*(z^4-10*z^2+9)*GFL(z;1)+(-107+10*z^2))/2700

%e GFL(z;4) = (1260*(z^6-35*z^4+259*z^2-225)*GFL(z;1)+(59845-7497*z^2+210*z^4))/ 1984500

%p nmax := 12; mmax := nmax: for n from 0 to nmax do cfn2(n, 0) := 1: cfn2(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax 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: for n from 1 to nmax do Delta(n-1) := sum((1-2^(2*k1-1))*(-1)^(n+1)*(-bernoulli(2*k1)/(2*k1))*(-1)^(k1+n)*cfn2(n-1, n-k1), k1=1..n)/ (2*4^(n-1)*(2*n-1)!) od: for n from 1 to nmax do LG1[ -2, n] := (-1)^(n+1)*4*Delta(n-1)* 4^(2*n-2)/binomial(2*n-2, n-1) od: for n from 1 to nmax do LGx[ -2, n] := LG1[ -2, n] od: for m from 0 to mmax do LGx[2*m, 1] := 2 od: for n from 2 to nmax do for m from 0 to mmax do LGx[2*m, n] := LGx[2*m-2, n-1]/((2*n-3)*(2*n-1)) - (2*n-3)*LGx[2*m, n-1]/(2*n-1) od: od: for n from 2 to nmax do s(n) := 0; for m from 0 to mmax-1 do s(n) := s(n) + LGx[2*m, n] od: od: seq(s(n), n=2..nmax);

%p # End program 1

%p nmax1:=5; ncol:=3; Digits:=20: mmax1:=nmax1: 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: for m from 1 to mmax1 do LG1[ -2*m, 1] := (((2^(2*m-1)-1)*bernoulli(2*m)/m)) od: LG1[0, 1] := evalf(gamma): for m from 2 to mmax1 do LG1[2*m-2, 1] := evalf(2*(1-2^(-2*m+1))*Zeta(2*m-1)) od: for m from -mmax1+ncol-1 to mmax1-1 do LG1[2*m, ncol] := sum((-1)^(k1+1)*cfn2(ncol-1, k1-1)* LG1[2*m-(2*ncol-2*k1), 1], k1=1..ncol)/(doublefactorial(2*ncol-3)*doublefactorial(2*ncol-1)) od;

%p # End program 2

%p # Maple programs edited by _Johannes W. Meijer_, Sep 25 2012

%Y See A162449 for the denominators of the column sums.

%Y The LAMBDA(z, n) polynomials and the LS1 matrix lead to the Lambda triangle A160487.

%Y The CFN2(z, n), the cfn2(n, k) and the LG2 matrix lead to A008956.

%Y The pg(n) and hg(n) sequences lead to A160476.

%Y The LG1[ -2, n] lead to A002197, A002198, A061549 and A001790.

%Y Cf. A001620 (gamma) and A079484 ((2n-1)!!*(2n+1)!!).

%Y Cf. A162440 (EG1 matrix), A162443 (BG1 matrix) and A162446 (ZG1 matrix)

%K easy,frac,sign

%O 2,1

%A _Johannes W. Meijer_, Jul 06 2009