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 A160480 The Beta triangle read by rows. 13
 -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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS 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. LINKS 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. J. M. Amigo, Relations among Sums of Reciprocal Powers Part II, International Journal of Mathematics and Mathematical Sciences , Volume 2008 (2008), pp. 1-20. Johannes W. Meijer, The zeros of the Eta, Zeta, Beta and Lambda polynomials, jpg and pdf, Mar 03 2013. FORMULA 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)!. EXAMPLE 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 MAPLE 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 MATHEMATICA 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; Table[BETA[n, m], {n, 2, 9}, {m, 1, n - 1}] // Flatten (* Jean-François Alcover, Dec 13 2017 *) CROSSREFS A160481 equals the rows sums . A101269 and A160482 equal the first and second left hand columns. A160483 and A160484 equal the second and third right hand columns. A160485 and A160486 are two related triangles. The CFN2(z, n) and the cfn2(n, k) lead to A008956. Cf. The Eta, Zeta and Lambda triangles A160464, A160474 and A160487. Cf. A162443 (BG1 matrix) Sequence in context: A330077 A132098 A223513 * A045998 A287194 A283083 Adjacent sequences:  A160477 A160478 A160479 * A160481 A160482 A160483 KEYWORD easy,sign,tabl AUTHOR Johannes W. Meijer, May 24 2009, Sep 19 2012 STATUS approved

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Last modified September 20 19:15 EDT 2020. Contains 337265 sequences. (Running on oeis4.)