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%I A160480
%S A160480 1,11,1,299,36,1,15371,2063,85,1,1285371,182474,8948,166,1,
%T A160480 159158691,23364725,1265182,29034,287,1,27376820379,4107797216,
%U A160480 237180483,6171928,77537,456,1
%V A160480 -1,-11,1,-299,36,-1,-15371,2063,-85,1,-1285371,182474,-8948,166,-1,
%W A160480 -159158691,23364725,-1265182,29034,-287,1,-27376820379,4107797216,
%X A160480 -237180483,6171928,-77537,456,-1
%N A160480 The Beta triangle read by rows.
%C A160480 The coefficients of the BS1 matrix are defined by BS1[2*m-1,n] = int(y^(2*m-1)/
(cosh(y))^(2*n-1),y=0..infinity)/factorial(2*m-1) for m = 1, 2, ..
and n = 1, 2, .. .
%C A160480 This definition leads to BS1[2*m-1,n=1] = 2*beta(2*m), for m = 1, 2,
.. , and the recurrence relation BS1 [2*m-1,n] = (2*n-3)/(2*n-2)*(BS1[2*m-1,
n-1]- BS1[2*m-3,n-1]/(2*n-3)^2) which we used to extend our definition
of the BS1 matrix coefficients to m = 0, -1, -2, .. . We discovered
that BS1[ -1,n] = 1 for n = 1, 2, .. . As usual beta(m) = sum((-1)^k/
(1+2*k)^m, k=0..infinity).
%C A160480 The coefficients in the columns of the BS1 matrix, for m = 1, 2, 3, ..
, and n = 2, 3, 4 .. , can be generated with the GK(z;n) polynomials
for which we found the following general expression GK(z;n) = ((-1)^(n+1)*CFN2(z;
n)*GK(z;n=1) + BETA(z;n))/p(n).
%C A160480 The CFN2(z;n) polynomials depend on the central factorial numbers A008956.
%C A160480 The BETA(z;n) are the Beta polynomials which lead to the Beta triangle.
%C A160480 The zero patterns of the Beta polynomials resemble a UFO. These patterns
resemble those of the Eta, Zeta and Lambda polynomials, see A160464,
A160474 and A160487.
%C A160480 The first Maple algorithm generates the coefficients of the Beta triangle.
The second Maple algorithm generates the BS1[2*m-1,n] coefficients
for m= 0, -1, -2, -3, .. .
%C A160480 Some of our results are conjectures based on numerical evidence, see
especially A160481.
%H A160480 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.
%H A160480 J.M. Amigo, Relations among Sums of Reciprocal Powers Part II, International Journal of Mathematics and Mathematical Sciences
, Volume 2008 (2008), pp. 1-20.
%F A160480 We discovered a relation between the Beta triangle coefficients BETA(n,
m) = (2*n-3)^2* BETA(n-1,m)- BETA(n-1,m-1) for n = 3, 4, .. and m
= 2, 3, .. with BETA(n,m=1) = (2*n-3)^2* BETA(n-1,m=1)-(2*n-4)! for
n = 2, 3, .. and BETA(n,n) = 0 for n = 1, 2, .. .
%F A160480 The generating functions GK(z;n) of the coefficients in the matrix columns
are defined by
%F A160480 GK(z;n) = sum(BS1[2*m-1,n]*z^(2*m-2), m=1..infinity) with n = 1, 2, ..
.
%F A160480 This definition leads to GK(z;n=1) = 1/(z*cos(Pi*z/2))*int(sin(z*t)/sin(t),
t=0..Pi/2).
%F A160480 Furthermore we discovered that GK(z;n) = GK(z;n-1)*((2*n-3)/(2*n-2)-z^2/
((2*n-2)*(2*n-3)))-1/((2*n-2)*(2*n-3)) for n = 2, 3 , .. .
%F A160480 We found the following general expression for the GK(z;n) polynomials,
for n = 2, 3, .. ,
%F A160480 GK(z;n) = ((-1)^(n+1)*CFN2(z;n)*GK(z;n=1) + BETA(z;n))/p(n) with p(n)
= (2*n-2)!.
%e A160480 The first few rows of the triangle BETA(n,m) with n=2,3,.. and m=1,2,
.. are
%e A160480 [ -1]
%e A160480 [ -11, 1]
%e A160480 [ -299, 36, -1]
%e A160480 [ -15371, 2063 -85, 1]
%e A160480 The first few BETA(z;n) polynomials are
%e A160480 BETA(z;n=2) = -1
%e A160480 BETA(z;n=3) = -11 + z^2
%e A160480 BETA(z;n=4) = -299 + 36*z^2 - z^4
%e A160480 The first few CFN1(z;n) polynomials are
%e A160480 CFN2(z;n=2) = (z^2-1)
%e A160480 CFN2(z;n=3) = (z^4-10*z^2+9)
%e A160480 CFN2(z;n=4) = (z^6- 35*z^4+259*z^2-225)
%e A160480 The first few generating functions GK(z;n) are:
%e A160480 GK(z;n=2) = ((-1)*(z^2-1)*GK(z,n=1) + (-1))/2
%e A160480 GK(z;n=3) = ((z^4-10*z^2+9)*GK(z,n=1)+ (-11 + z^2))/24
%e A160480 GK(z;n=4) = ((-1)*( z^6- 35*z^4+259*z^2-225)*GK(z,n=1) + (-299 + 36*z^2
- z^4))/720
%p A160480 restart; nmax:=8; mmax:=nmax: for n from 1 to nmax do BETA(n,n):=0 end
do: m:=1: for n from m+1 to nmax do BETA(n,m):=(2*n-3)^2*BETA(n-1,
m)-(2*n-4)! od: for m from 2 to mmax do for n from m+1 to nmax do
BETA(n,m):=(2*n-3)^2*BETA(n-1,m)-BETA(n-1,m-1) od: od: for n from
2 to nmax do for m from 1 to n-1 do b((((n-2)*(n-1)/2))+m+1):=BETA(n,
m) end do end do: a:=n-> b(n): seq(a(n), n=2..(nmax-1)*(nmax)/2+1);
%p A160480 restart; nmax:=10; m:=1; BS1row:=1-2*m; jn:=nmax: im:=nmax: for n from
1 to nmax do for i from 2 to im do cfn2[i,1]:=0 end do: for j from
1 to jn do cfn2[1,j]:=1 end do: for j from 2 to jn do for i from
2 to im do cfn2[i,j]:= cfn2[i,j-1] + cfn2[i-1,j-1]*(2*j-3)^2 end
do end do: end do: mmax:=nmax: for m1 from 1 to mmax do BS1[1-2*m1,
1]:= euler(2*m1-2) od: for n from 2 to nmax do for m1 from 1 to mmax-n+1
do BS1[1-2*m1,n]:= (-1)^(n+1)*sum((-1)^(k+1)*cfn2[k,n]* BS1[2*k-2*n-2*m1+1,
1],k=1..n)/(2*n-2)! od: od: a:=n-> BS1[1-2*m,n]: seq(a(n), n=1..nmax-m+1);
%Y A160480 A160481 equals the rows sums .
%Y A160480 A101269 and A160482 equal the first and second left hand columns.
%Y A160480 A160483 and A160484 equal the second and third right hand columns.
%Y A160480 A160485 and A160486 are two related triangles.
%Y A160480 The CFN2(z, n) and the cfn2(m, n) lead to A008956.
%Y A160480 Cf. The Eta, Zeta and Lambda triangles A160464, A160474 and A160487.
%Y A160480 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06
2009: (Start)
%Y A160480 Cf. A162443 (BG1 matrix).
%Y A160480 (End)
%Y A160480 Sequence in context: A038315 A093158 A132098 this_sequence A045998 A027645
A010190
%Y A160480 Adjacent sequences: A160477 A160478 A160479 this_sequence A160481 A160482
A160483
%K A160480 easy,sign,tabl
%O A160480 2,2
%A A160480 Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009, Jun 29 2009
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