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 A162005 The EG1 triangle. 16

%I

%S 1,2,1,16,28,1,272,1032,270,1,7936,52736,36096,2456,1,353792,3646208,

%T 4766048,1035088,22138,1,22368256,330545664,704357760,319830400,

%U 27426960,199284,1,1903757312,38188155904,120536980224,93989648000

%N The EG1 triangle.

%C 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.

%C The n-th 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.

%C 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.

%C 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, .. .

%C Contribution from _Johannes W. Meijer_, Nov 23 2009: (Start)

%C 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).

%C (End)

%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 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).

%e The first few rows of the EG1 triangle are :

%e [1]

%e [2, 1]

%e [16, 28, 1]

%e [272, 1032, 270, 1]

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

%e RG(z,-1) = 1

%e RG(z,-3) = 2+z

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

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

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

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

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

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

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

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

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

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

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

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

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

%e ECGP(-1,n) = 1

%e ECGP(-3,n) = n

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

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

%p nmax:=7; mmax := nmax: imax := nmax: 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(j1)*(x+1)^(j1), j1=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))), ascending) od: for n from 1 to nmax do for m from 1 to n do a(n, m) := abs(coeff(g(n), z, m-1)) od: od: seq(seq(a(n, m), m=1..n), n=1..nmax);

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

%Y A079484 equals the row sums.

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

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

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

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

%Y Cf. A167560 (ED2 array).

%K easy,nonn,tabl

%O 1,2

%A _Johannes W. Meijer_, Jun 27 2009, Jul 02 2009, Aug 31 2009

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Last modified August 18 06:34 EDT 2018. Contains 313823 sequences. (Running on oeis4.)