2,1

The EG1 matrix coefficients are defined by EG1[2m-1,1] = 2*eta(2m-1) and the recurrence relation EG1[2m-1,n] = EG1[2m-1,n-1] - EG1[2m-3,n-1]/(n-1)^2 with m = .. , -2, -1, 0, 1, 2, .. and n = 1, 2, 3, .. . As usual eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function. For the EG2 matrix, the even counterpart of the EG1 matrix, see A008955.

The coefficients in the columns of the EG1 matrix, for m => 1 and n => 2, can be generated with GFE(z;n) = ((-1)^(n-1)*r(n)*CFN1(z,n)*GFE(z;n=1) + ETA(z,n))/pg(n) for n => 2.

The CFN1(z,n) polynomials depend on the central factorial numbers A008955 and the ETA(z,n) are the Eta polynomials which led to the Eta triangle, see for both A160464.

The pg(n) sequence can be generated with the first Maple program and the EG1[2m-1,n] matrix coefficients can be generated with the second Maple program.

The EG1 matrix is related to the ES1 matrix, see A160464 and the formulae below.

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.

pg(n) = (n-1)!^2*2^floor(ln(n-1)/ln(2)+1) for n => 2.

r(n) = 2^e(n) = 2^floor(log(n-1)/log(2)+1) for n => 2.

EG1[ -1,n] = 2^(1-2*n)*(2*n-1)!/((n-1)!^2) for n=>1.

GFE(z;n) = sum (EG1[2*m-1,n]*z^(2*m-2),m=1..infinity)

GFE(z;n) = (1-z^2/(n-1)^2)*GFE(z;n-1)-EG1[ -1,n-1]/(n-1)^2 for n =>2 with GFE(z;n=1) = 2*ln(2)-Psi(z)-Psi(-z)+Psi(z/2)+Psi(-z/2) and Psi(z) is the digamma function.

EG1[2m-1,n] = (2*2^(1-2*n)*(2*n-1)!/((n-1)!^2)) * ES1[2m-1,n]

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

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

GFE(z;n=3) = ((+1)*4*(z^4 - 5*z^2 + 4) *GFE(z;n=1) + (-11 + 2*z^2))/16

GFE(z;n=4) = ((-1)*4*(z^6-14*z^4+49*z^2-36)*GFE(z;n=1) + (-114+29*z^2-2*z^4))/144

restart; nmax:=16; seq((n-1)!^2*2^floor(ln(n-1)/ln(2)+1), n=2..nmax);

restart; nmax:=5; coln:=4; mmax:=nmax: for n from 0 to nmax do t1(n, 0):=1 end do: for n from 0 to nmax do t1(n, n):=(n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do t1(n, m):= t1(n-1, m-1)*n^2+t1(n-1, m) end do: end do: T:=0: for n from 0 to nmax do for m from 0 to n do a(T):=t1(n, m): T:=T+1 od: od: for m from 1 to mmax do EG1[1-2*m, 1]:=evalf((2^(2*m)-1)* bernoulli(2*m)/(m)) od: EG1[1, 1]:=evalf(2*ln(2)): for m from 2 to mmax do EG1[2*m-1, 1]:= evalf(2*(1-2^(1-(2*m-1)))*Zeta(2*m-1)) od: for m from -mmax+coln to mmax do EG1[2*m-1, coln]:= (-1)^(coln+1)*sum((-1)^k*t1(coln-1, k)*EG1[1-2*coln+2*m+2*k, 1], k=0..coln-1)/(coln-1)!^2 od;

The ETA(z, n) polynomials and the ES1 matrix lead to the Eta triangle A160464.

The CFN1(z, n), the t1(n, m) and the EG2 matrix lead to A008955.

The EG1[ -1, n] equal (1/2)*A001803(n-1)/A046161(n-1).

The r(n) sequence equals A062383(n) (n=>1).

The e(n) sequence equals A029837(n) (n=>1).

Cf. A160473 (p(n) sequence).

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

Sequence in context: A151402 A003768 A024915 this_sequence A103885 A124578 A085510

Adjacent sequences: A162437 A162438 A162439 this_sequence A162441 A162442 A162443

easy,nonn

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