%I A162005
%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 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.
%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 (meijgia(AT)hotmail.com), 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)
%H A162005 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
abramowitz_and_stegun/">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 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:
%p A162005 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);
%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 Sequence in context: A016447 A095850 A113108 this_sequence A013125 A012967
A013121
%Y A162005 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 23
2009: (Start)
%Y A162005 Cf. A167560 (ED2 array).
%Y A162005 (End)
%Y A162005 Adjacent sequences: A162002 A162003 A162004 this_sequence A162006 A162007
A162008
%K A162005 easy,nonn,tabl,new
%O A162005 1,2
%A A162005 Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 27 2009, Jul 02 2009,
Aug 31 2009
%E A162005 Maple program fixed by Johannes W. Meijer (meijgia(AT)hotmail.com), Oct
06 2009
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