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A162005 The EG1 triangle.

%I A162005 #16 Apr 13 2019 16:15:00
%S A162005 1,2,1,16,28,1,272,1032,270,1,7936,52736,36096,2456,1,353792,3646208,
%T A162005 4766048,1035088,22138,1,22368256,330545664,704357760,319830400,
%U A162005 27426960,199284,1,1903757312,38188155904,120536980224,93989648000
%N A162005 The EG1 triangle.
%C A162005 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 A162005 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 A162005 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 A162005 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 A162005 Contribution from _Johannes W. Meijer_, Nov 23 2009: (Start)
%C A162005 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 A162005 (End)
%C A162005 Appears to equal triangle A322230 with rows read in reverse order. Triangle A322230 describes the e.g.f. S(x,k) = Integral C(x,k)*D(x,k)^2 dx, such that C(x,k)^2 - S(x,k)^2 = 1, and D(x,k)^2 - k^2*S(x,k)^2 = 1. - _Paul D. Hanna_, Dec 22 2018
%C A162005 Appears to equal triangle A325220, which has e.g.f. S(x,k) = -i * sn( i * Integral C(x,k) dx, k) such that C(x,k) = cn( i * Integral C(x,k) dx, k), where sn(x,k) and cn(x,k) are Jacobi Elliptic functions. - _Paul D. Hanna_, Apr 13 2019
%H A162005 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 A162005 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 A162005 The first few rows of the EG1 triangle are :
%e A162005 [1]
%e A162005 [2, 1]
%e A162005 [16, 28, 1]
%e A162005 [272, 1032, 270, 1]
%e A162005 The first few RG(z,1-2*m) polynomials are:
%e A162005 RG(z,-1) = 1
%e A162005 RG(z,-3) = 2+z
%e A162005 RG(z,-5) = 16+28*z+z^2
%e A162005 RG(z,-7) = 272+1032*z+270*z^2+z^3
%e A162005 The first few GFREG1(z,1-2*m) are:
%e A162005 GFREG1(z,-1) = (1)*(1)/(2*(1-z)^(3/2))
%e A162005 GFREG1(z,-3) = (-1)*(2+z)/(2^3*(1-z)^(5/2))
%e A162005 GFREG1(z,-5) = (1)*(16+28*z+z^2)/( 2^5*(1-z)^(7/2))
%e A162005 GFREG1(z,-7) = (-1)*(272+1032*z+270*z^2+z^3)/(2^7*(1-z)^(9/2))
%e A162005 The first few REG1(1-2*m,n) are:
%e A162005 REG1(-1,n) = (1/1)*(1)*(1/n)*4^(-n)*(2*n)!/(n-1)!^2
%e A162005 REG1(-3,n) = (-1/2)*(n) *(1/n)*4^(-n)*(2*n)!/(n-1)!^2
%e A162005 REG1(-5,n) = (1/4) *(n+3*n^2) *(1/n)*4^(-n)*(2*n)!/(n-1)!^2
%e A162005 REG1(-7,n) = (-1/8)*(4*n+15*n^2+15*n^3) *(1/n)*4^(-n)*(2*n)!/(n-1)!^2
%e A162005 The first few ECGP(1-2*m,n) polynomials are:
%e A162005 ECGP(-1,n) = 1
%e A162005 ECGP(-3,n) = n
%e A162005 ECGP(-5,n) = n+3*n^2
%e A162005 ECGP(-7,n) = 4*n+15*n^2+15*n^3
%p A162005 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 A162005 # Maple program edited by _Johannes W. Meijer_, Sep 25 2012
%Y A162005 A079484 equals the row sums.
%Y A162005 A000182 (ZAG numbers), A162006 and A162007 equal the first three left hand columns.
%Y A162005 A000012, A004004 (2x), A162008, A162009 and A162010 equal the first five right hand columns.
%Y A162005 Related to A094665, A083061 and A156919 (DEF triangle).
%Y A162005 Cf. A161198 [(1-x)^((-1-2*n)/2)], A008955 (EG2[2m, n])
%Y A162005 Cf. A167560 (ED2 array).
%Y A162005 Cf. A322230 (reversed rows), A325220.
%K A162005 easy,nonn,tabl
%O A162005 1,2
%A A162005 _Johannes W. Meijer_, Jun 27 2009, Jul 02 2009, Aug 31 2009