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A160480
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The Beta triangle read by rows.
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13
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-1, -11, 1, -299, 36, -1, -15371, 2063, -85, 1, -1285371, 182474, -8948, 166, -1, -159158691, 23364725, -1265182, 29034, -287, 1, -27376820379, 4107797216, -237180483, 6171928, -77537, 456, -1
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OFFSET
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2,2
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COMMENTS
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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, ... .
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).
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).
The CFN2(z;n) polynomials depend on the central factorial numbers A008956.
The BETA(z;n) are the Beta polynomials which lead to the Beta triangle.
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.
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, ... .
Some of our results are conjectures based on numerical evidence, see especially A160481.
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LINKS
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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.
Johannes W. Meijer, The zeros of the Eta, Zeta, Beta and Lambda polynomials, jpg and pdf, Mar 03 2013.
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FORMULA
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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, ... .
The generating functions GK(z;n) of the coefficients in the matrix columns are defined by
GK(z;n) = sum(BS1[2*m-1,n]*z^(2*m-2), m=1..infinity) with n = 1, 2, ... .
This definition leads to GK(z;n=1) = 1/(z*cos(Pi*z/2))*int(sin(z*t)/sin(t),t=0..Pi/2).
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, ... .
We found the following general expression for the GK(z;n) polynomials, for n = 2, 3, ...,
GK(z;n) = ((-1)^(n+1)*CFN2(z;n)*GK(z;n=1) + BETA(z;n))/p(n) with p(n) = (2*n-2)!.
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EXAMPLE
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The first few rows of the triangle BETA(n,m) with n=2,3,... and m=1,2,... are
[ -1],
[ -11, 1],
[ -299, 36, -1],
[ -15371, 2063 -85, 1].
The first few BETA(z;n) polynomials are
BETA(z;n=2) = -1,
BETA(z;n=3) = -11 + z^2,
BETA(z;n=4) = -299 + 36*z^2 - z^4.
The first few CFN1(z;n) polynomials are
CFN2(z;n=2) = (z^2 - 1),
CFN2(z;n=3) = (z^4 - 10*z^2 + 9),
CFN2(z;n=4) = (z^6 - 35*z^4 + 259*z^2 - 225).
The first few generating functions GK(z;n) are
GK(z;n=2) = ((-1)*(z^2-1)*GK(z,n=1) + (-1))/2,
GK(z;n=3) = ((z^4 - 10*z^2 + 9)*GK(z,n=1)+ (-11 + z^2))/24,
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.
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MAPLE
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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: seq(seq(BETA(n, m), m=1..n-1), n= 2..nmax);
# End first program
nmax1 := 25; m := 1; BS1row := 1-2*m; for n from 0 to nmax1 do cfn2(n, 0) := 1: cfn2(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax1 do for k from 1 to n-1 do cfn2(n, k) := (2*n-1)^2*cfn2(n-1, k-1) + cfn2(n-1, k) od: od: mmax1 := nmax1: for m1 from 1 to mmax1 do BS1[1-2*m1, 1] := euler(2*m1-2) od: for n from 2 to nmax1 do for m1 from 1 to mmax1-n+1 do BS1[1-2*m1, n] := (-1)^(n+1)*sum((-1)^(k1+1)*cfn2(n-1, k1-1) * BS1[2*k1-2*n-2*m1+1, 1], k1 =1..n)/(2*n-2)! od: od: seq(BS1[1-2*m, n], n=1..nmax1-m+1);
# End second program
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MATHEMATICA
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BETA[2, 1] = -1;
BETA[n_, 1] := BETA[n, 1] = (2*n - 3)^2*BETA[n - 1, 1] - (2*n - 4)!;
BETA[n_ /; n > 2, m_ /; m > 0] /; 1 <= m <= n := BETA[n, m] = (2*n - 3)^2*BETA[n - 1, m] - BETA[n - 1, m - 1];
BETA[_, _] = 0;
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CROSSREFS
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The CFN2(z, n) and the cfn2(n, k) lead to A008956.
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KEYWORD
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AUTHOR
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STATUS
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approved
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