1,2

We define the EG1 matrix 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 for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. , with 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[2m,n] coefficients see A008955.

The nth term of the row coefficients EG1[1-2*m,n] for m = 1, 2, .. , can be generated with REG1(1-2*m,n) = (-1)^(m+1)*2^(1-m)*ECGP(1-2*m, n)*(1/n)*4^(-n)*(2*n)!/((n-1)!)^2 . For information about the ECGP polynomials see A094665 and the examples below.

We define the o.g.f.s. of the REG1(1-2*m,n) by GFREG1(z,1-2*m) = sum(REG1(1-2*m,n)* z^(n-1), n=1..infinity) for m = 1, 2, .. , with GFREG1(z,1-2*m) = (-1)^(m+1)* RG(z,1-2*m)/ (2^(2*m-1)*(1-z)^((2*m+1)/2)). The RG(z,1-2m) polynomials led to the EG1 triangle.

We used the coefficients of the A156919 and A094665 triangles to determine those of the EG1 triangle, see the Maple program. The A156919 triangle gives information about the sums SF(p) = sum(n^(p-1)*4^(-n)*z^(n-1)*(2*n)!/((n-1)!)^2, n=1..infinity) for p= 0, 1, 2, .. .

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 23 2009: (Start)

The EG1 matrix is related to the ED2 array A167560 because sum(EG1(2*m-1,n)*z^(2*m-1), m=1..infinity) = ((2*n-1)!/(4^(n-1)*(n-1)!^2))*int(sinh(y*(2*z))/cosh(y)^(2*n),y=0..infinity).

(End)

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 different form of the recurrence relation is EG1[1-2*m,n] = (EG1[3-2*m,n]-EG1[3-2*m,n+1])* (n^2) for m = 2, 3, .. , with EG1[ -1,n] = (1/n)*4^(-n)*((2*n)!/(n-1)!^2).

The first few rows of the EG1 triangle are :

[1]

[2, 1]

[16, 28, 1]

[272, 1032, 270, 1]

The first few RG(z,1-2*m) polynomials are:

RG(z,-1) = 1

RG(z,-3) = 2+z

RG(z,-5) = 16+28*z+z^2

RG(z,-7) = 272+1032*z+270*z^2+z^3

The first few GFREG1(z,1-2*m) are:

GFREG1(z,-1) = (1)*(1)/(2*(1-z)^(3/2))

GFREG1(z,-3) = (-1)*(2+z)/(2^3*(1-z)^(5/2))

GFREG1(z,-5) = (1)*(16+28*z+z^2)/( 2^5*(1-z)^(7/2))

GFREG1(z,-7) = (-1)*(272+1032*z+270*z^2+z^3)/(2^7*(1-z)^(9/2))

The first few REG1(1-2*m,n) are:

REG1(-1,n) = (1/1)*(1)*(1/n)*4^(-n)*(2*n)!/(n-1)!^2

REG1(-3,n) = (-1/2)*(n) *(1/n)*4^(-n)*(2*n)!/(n-1)!^2

REG1(-5,n) = (1/4) *(n+3*n^2) *(1/n)*4^(-n)*(2*n)!/(n-1)!^2

REG1(-7,n) = (-1/8)*(4*n+15*n^2+15*n^3) *(1/n)*4^(-n)*(2*n)!/(n-1)!^2

The first few ECGP(1-2*m,n) polynomials are:

ECGP(-1,n) = 1

ECGP(-3,n) = n

ECGP(-5,n) = n+3*n^2

ECGP(-7,n) = 4*n+15*n^2+15*n^3

restart; nmax:=8; mmax:=nmax: imax := nmax: i:=0: T1(0, x):=1: T1(0, x+1):=1: for i from 1 to imax do T1(i, x):= expand((2*x+1)*(x+1)*T1(i-1, x+1)-2*x^2*T1(i-1, x)): dx:=degree(T1(i, x)): for k from 0 to dx do c(k):=coeff(T1(i, x), x, k) od: T1(i, x+1):=sum(c(j)*(x+1)^(j), j=0..dx): od: for i from 0 to imax do for j from 0 to i do A083061(i, j):=coeff(T1(i, x), x, j) od: od: for n from 0 to nmax do for k from 0 to n do A094665(n+1, k+1) := A083061(n, k) od: od: A094665(0, 0):=1: for n from 1 to nmax do A094665(n, 0):=0 od:

for m from 1 to mmax do A156919(0, m):=0 end do: for n from 0 to nmax do A156919(n, 0):=2^n end do: for n from 1 to nmax do for m from 1 to mmax do A156919(n, m):=(2*m+2)*A156919(n-1, m)+(2*n-2*m+1)* A156919(n-1, m-1) end do end do: for n from 0 to nmax do SF(n):= sum(A156919(n, k1)*z^k1, k1=0..n)/(2^(n+1)*(1-z)^((2*n+3)/2)) od: GFREG1(z, -1):=A156919(0, 0)*A094665 (0, 0)/ (2*(1-z)^(3/2)): for m from 2 to nmax do GFREG1(z, 1-2*m):= simplify((-1)^(m+1)*2^(1-m)* sum(A094665(m-1, k2)*SF(k2), k2=1..m-1)) od: for m from 1 to mmax do g(m):= sort((numer ((-1)^(m+1)* GFREG1(z, 1-2*m))), z, ascending) od: T:=0: for n from 1 to nmax do for m from 1 to n do T:=T+1: a(n, m):=coeff(g(n), z, m-1): a(T):=a(n, m) od: od: seq(a(n), n=1..T);

A079484 equals the row sums.

A000182 (ZAG numbers), A162006 and A162007 equal the first three left hand columns.

A000012, A004004 (2x), A162008, A162009 and A162010 equal the first five right hand columns.

Related to A094665, A083061 and A156919 (DEF triangle).

Cf. A161198 [(1-x)^((-1-2*n)/2)], A008955 (EG2[2m, n])

Sequence in context: A016447 A095850 A113108 this_sequence A013125 A012967 A013121

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 23 2009: (Start)

Cf. A167560 (ED2 array).

(End)

Adjacent sequences: A162002 A162003 A162004 this_sequence A162006 A162007 A162008

easy,nonn,tabl,new

Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 27 2009, Jul 02 2009, Aug 31 2009

Maple program fixed by Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 06 2009