A160480 -id:A160480 - cp's OEIS frontend

A160481 Row sums of the Beta triangle A160480.

-1, -10, -264, -13392, -1111680, -137030400, -23500108800, -5351202662400, -1562069156659200, -568747270103040000, -252681700853514240000, -134539938778433126400000, -84573370199475510312960000, -61972704966344777143418880000, -52361960516341326660973363200000

Offset: 2

Johannes W. Meijer, May 24 2009, Sep 19 2012

It is conjectured that the row sums of the Beta triangle depend on three different sequences. Two Maple algorithms are given. The first one gives the row sums according to the Beta triangle A160480 and the second one gives the row sums according to our conjecture.

Christopher P. Herzog, Kuo-Wei Huang, and Kristan Jensen, Universal Entanglement and Boundary Geometry in Conformal Field Theory, arXiv preprint arXiv:1510.00021 [hep-th], 2015.
Kuo-Wei Huang, Resummation of Multi-Stress Tensors in Higher Dimensions, arXiv:2406.07458 [hep-th], 2024. See p. 10.

A160480 is the Beta triangle.
Row sum factors A120778, A000165 and A049606.
Cf. A002474, A009445, A129890.

nmax := 14; 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 s1(n) := 0: for m from 1 to n-1 do s1(n) := s1(n) + BETA(n, m) od: od: seq(s1(n), n=2..nmax);
# End first program
nmax := nmax; A120778 := proc(n): numer(sum(binomial(2*k1, k1)/(k1+1) / 4^k1, k1=0..n)) end proc: A000165 := proc(n): 2^n*n! end proc: A049606 := proc(n): denom(2^n/n!) end proc: for n from 2 to nmax do s2(n) := (-1)*A120778(n-2)*A000165(n-2)*A049606(n-1) end do: seq(s2(n), n=2..nmax);
# End second program

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[Sum[BETA[n, m], {m, 1, n - 1}], {n, 2, 14}] (* Jean-François Alcover, Dec 13 2017 *)

Rowsums(n) = (-1)*A120778(n-2)*A000165(n-2)*A049606(n-1) for n >= 2.
Conjecture: a(n) = (2*n-3)! - 2^(2*n-3)*(n-1)!*(n-2)!, for n >= 2 (gives the first 13 terms). - Christopher P. Herzog, Nov 25 2014
Meijer's and Herzog's conjectures can also be written as: a(n) = -A129890(n-2)*A000165(n-2) = A009445(n-2) - A002474(n-2). - Peter Luschny, Dec 01 2014

a(15)-a(16) from Stefano Spezia, Jun 28 2024

A160482 Second left hand column of the Beta triangle A160480.

Original entry on oeis.org

1, 36, 2063, 182474, 23364725, 4107797216, 951631193979, 281268377936406, 103364179496231841, 46249298048633298156, 24761883016575811181799, 15633887186628672922202850, 11496293145747030133211784525

Offset: 3

Johannes W. Meijer, May 24 2009

A160480 is the Beta triangle.

A101269 is the first left hand column.

nmax:=15; a(1,1):=0: for n from 2 to nmax do a(n,1):=(2*n-3)^2*a(n-1,1)-(2*(n-2))! od: a(2,2):=0: for n from 3 to nmax do a(n,2):=(2*n-3)^2*a(n-1,2)-a(n-1,1) od: seq(a(n,2), n=3..nmax);

A160483 Second right hand column of the Beta triangle A160480.

Original entry on oeis.org

-11, 36, -85, 166, -287, 456, -681, 970, -1331, 1772, -2301, 2926, -3655, 4496, -5457, 6546, -7771, 9140, -10661, 12342, -14191, 16216, -18425, 20826, -23427, 26236, -29261, 32510, -35991, 39712, -43681, 47906, -52395, 57156

Offset: 1

Johannes W. Meijer, May 24 2009

Index entries for linear recurrences with constant coefficients, signature (-4,-6,-4,-1).

A160480 is the Beta triangle.

A160484 is the third right hand column.

a(n) = (-1)^(n)*(2*n+1)^2 - a(n-1) for n = 2, 3, .. , with a(1) = -11.
From Chai Wah Wu, Sep 24 2020: (Start)
a(n) = - 4*a(n-1) - 6*a(n-2) - 4*a(n-3) - a(n-4) for n > 4.
G.f.: x*(-2*x^3 - 7*x^2 - 8*x - 11)/(x + 1)^4. (End)

A160484 Third right hand column of the Beta triangle A160480.

Original entry on oeis.org

-299, 2063, -8948, 29034, -77537, 180137, -376946, 727116, -1314087, 2251475, -3689600, 5822654, -8896509, 13217165, -19159838, 27178688, -37817187, 51719127, -69640268, 92460626, -121197401, 157018545, -201256970, 255425396

Offset: 1

Johannes W. Meijer, May 24 2009

Index entries for linear recurrences with constant coefficients, signature (-7,-21,-35,-35,-21,-7,-1).

A160480 is the Beta triangle.

A160483 is the second right hand column.

nmax:=27; mmax:=nmax: for m from 1 to mmax do BETA(2,m):=0 end do: BETA(2,1):=-1: for n from 3 to nmax do BETA(n,1):=(2*n-3)^2*BETA(n-1,1)-(2*n-4)! end do: for n from 3 to nmax do for m from 2 to mmax do BETA(n,m):=(2*n-3)^2*BETA(n-1,m)-BETA(n-1,m-1) end do end do: for n from 4 to nmax do a(n-3):=BETA(n,n-3) od: seq(a(n), n=1..nmax-3);

LinearRecurrence[{-7,-21,-35,-35,-21,-7,-1},{-299,2063,-8948,29034,-77537,180137,-376946},30] (* Harvey P. Dale, Jul 23 2021 *)

From Chai Wah Wu, Sep 24 2020: (Start)
a(n) = - 7*a(n-1) - 21*a(n-2) - 35*a(n-3) - 35*a(n-4) - 21*a(n-5) - 7*a(n-6) - a(n-7) for n > 7.
G.f.: x*(-24*x^6 - 162*x^5 - 467*x^4 - 744*x^3 - 786*x^2 - 30*x - 299)/(x + 1)^7. (End)

A008956 Triangle of central factorial numbers |4^k t(2n+1,2n+1-2k)| read by rows (n>=0, k=0..n).

Original entry on oeis.org

1, 1, 1, 1, 10, 9, 1, 35, 259, 225, 1, 84, 1974, 12916, 11025, 1, 165, 8778, 172810, 1057221, 893025, 1, 286, 28743, 1234948, 21967231, 128816766, 108056025, 1, 455, 77077, 6092515, 230673443, 3841278805, 21878089479, 18261468225, 1, 680

Offset: 0

N. J. A. Sloane

The n-th row gives the coefficients in the expansion of Product_{i=0..n-1} (x+(2i+1)^2), highest powers first (see the discussion of central factorial numbers in A008955). - N. J. A. Sloane, Feb 01 2011
Descending row polynomials in x^2 evaluated at k generate odd coefficients of e.g.f. sin(arcsin(kt)/k): 1, x^2 - 1, 9x^4 - 10x^2 + 1, 225x^6 - 259x^4 + 34x^2 - 1, ... - Ralf Stephan, Jan 16 2005
From Johannes W. Meijer, Jun 18 2009: (Start)
We define (Pi/2)*Beta(n-1/2-z/2,n-1/2+z/2)/Beta(n-1/2,n-1/2) = (Pi/2)*Gamma(n-1/2-z/2)* Gamma(n-1/2+z/2)/Gamma(n-1/2)^2 = sum(BG2[2m,n]*z^(2m), m = 0..infinity) with Beta(z,w) the Beta function. Our definition leads to BG2[2m,1] = 2*beta(2m+1) and the recurrence relation BG2[2m,n] = BG2[2m,n-1] - BG2[2m-2,n-1]/(2*n-3)^2 for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. , with beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). We observe that beta(2m+1) = 0 for m = -1, -2, -3, .. .We found for the BG2[2*m,n] = sum((-1)^(k+n)*t2(n-1,k-1)* 2*beta(2*m-2*n+2*k+1),k=1..n)/((2*n-3)!!)^2 with the central factorial numbers t2(n,m) as defined above; see also the Maple program.
From the BG2 matrix and the closely related EG2 and ZG2 matrices, see A008955, we arrive at the LG2 matrix which is defined by LG2[2m-1,1] = 2*lambda(2*m) and the recurrence relation LG2[2*m-1,n] = LG2[2*m-3,n-1]/((2*n-3)*(2*n-1)) - (2*n-3)*LG2[2*m-1,n-1]/(2*n-1) for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. , with lambda(m) = (1-2^(-m))*zeta(m) with zeta(m) the Riemann zeta function. We found for the matrix coefficients LG2[2m-1,n] = sum((-1)^(k+1)* t2(n-1,k-1)*2*lambda(2*m-2*n+2*k)/((2*n-1)!!*(2*n-3)!!), k=1..n) and we see that the central factorial numbers t2(n,m) once again play a crucial role.
(End)

			Triangle begins:
[1]
[1, 1]
[1, 10, 9]
[1, 35, 259, 225]
[1, 84, 1974, 12916, 11025]
[1, 165, 8778, 172810, 1057221, 893025]
[1, 286, 28743, 1234948, 21967231, 128816766, 108056025]
[1, 455, 77077, 6092515, 230673443, 3841278805, 21878089479, 18261468225]
...

P. L. Butzer, M. Schmidt, E. L. Stark and L. Vogt, Central Factorial Numbers: Their main properties and some applications, Numerical Functional Analysis and Optimization, 10 (5&6), 419-488 (1989). [From Johannes W. Meijer, Jun 18 2009]
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
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. [From _Johannes W. Meijer_, Jun 18 2009]
R. H. Boels, T. Hansen, String theory in target space, arXiv preprint arXiv:1402.6356 [hep-th], 2014.
T. L. Curtright, D. B. Fairlie, C. K. Zachos, A compact formula for rotations as spin matrix polynomials, arXiv preprint arXiv:1402.3541 [math-ph], 2014.
T. L. Curtright, T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arXiv:1408.0767 [math-ph], 2014.
M. Eastwood and H. Goldschmidt, Zero-energy fields on complex projective space, arXiv preprint arXiv:1108.1602 [math.DG], 2011.
M. Eastwood, The X-ray transform on projective space. - From _N. J. A. Sloane_, Oct 22 2012

Cf. A008958.
Columns include A000447, A001823. Right-hand columns include A001818, A001824, A001825. Cf. A008955.
Appears in A160480 (Beta triangle), A160487 (Lambda triangle), A160479 (ZL(n) sequence), A161736, A002197 and A002198. - Johannes W. Meijer, Jun 18 2009
Cf. A162443 (BG1 matrix) and A162448 (LG1 matrix). - Johannes W. Meijer, Jul 06 2009
Cf. A001147.

a008956 n k = a008956_tabl !! n !! k
a008956_row n = a008956_tabl !! n
a008956_tabl = [1] : f [1] 1 1 where
   f xs u t = ys : f ys v (t * v) where
     ys = zipWith (+) (xs ++ [t^2]) ([0] ++ map (* u^2) (init xs) ++ [0])
     v = u + 2
-- Reinhard Zumkeller, Dec 24 2013

f:=n->mul(x+(2*i+1)^2,i=0..n-1);
for n from 0 to 12 do
t1:=eval(f(n)); t1d:=degree(t1);
t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
lprint(t2);
od: # N. J. A. Sloane, Feb 01 2011
A008956 := proc(n,k) local i ; mul( x+2*i-2*n-1,i=1..2*n) ; expand(%) ; coeftayl(%,x=0,2*(n-k)) ; abs(%) ; end: for n from 0 to 10 do for k from 0 to n do printf("%a,",A008956(n,k)) ; od: od: # R. J. Mathar, May 29 2009
nmax:=7: for n from 0 to nmax do t2(n, 0):=1: t2(n, n):=(doublefactorial(2*n-1))^2 od: for n from 1 to nmax do for k from 1 to n-1 do t2(n, k) := (2*n-1)^2*t2(n-1, k-1)+t2(n-1, k) od: od: seq(seq(t2(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jun 18 2009, Revised Sep 16 2012

t[, 0] = 1; t[n, n_] := t[n, n] = ((2*n-1)!!)^2; t[n_, k_] := t[n, k] = (2*n-1)^2*t[n-1, k-1] + t[n-1, k]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 07 2014, after Johannes W. Meijer *)

{T(n, k) = if( n<=0, k==0, (-1)^k * polcoeff( numerator( 2^(2*n -1) / sum(j=0, 2*n - 1, binomial( 2*n - 1, j) / (x + 2*n - 1 - 2*j))), 2*n - 2*k))}; /* Michael Somos, Feb 24 2003 */

Conjecture row sums: Sum_{k=0..n} T(n,k) = |A101927(n+1)|. - R. J. Mathar, May 29 2009

May be generated by the recurrence t2(n,k) = (2*n-1)^2*t2(n-1,k-1)+t2(n-1,k) with t2(n,0) = 1 and t2(n,n)=((2*n-1)!!)^2. - Johannes W. Meijer, Jun 18 2009

More terms from Vladeta Jovovic, Apr 16 2000

Edited by N. J. A. Sloane, Feb 01 2011

A160464 The Eta triangle.

Original entry on oeis.org

-1, -11, 2, -114, 29, -2, -3963, 1156, -122, 4, -104745, 32863, -4206, 222, -4, -3926745, 1287813, -184279, 12198, -366, 4, -198491580, 67029582, -10317484, 781981, -30132, 562, -4

Offset: 2

Johannes W. Meijer, May 24 2009

The ES1 matrix coefficients are defined by ES1[2*m-1,n] = 2^(2*m-1) * int(y^(2*m-1)/(cosh(y))^(2*n),y=0..infinity)/(2*m-1)! for m = 1, 2, 3, .. and n = 1, 2, 3 .. .
This definition leads to ES1[2*m-1,n=1] = 2*eta(2*m-1) and the recurrence relation ES1[2*m-1,n] = ((2*n-2)/(2*n-1))*(ES1[2*m-1,n-1] - ES1[2*m-3,n-1]/(n-1)^2) which we used to extend our definition of the ES1 matrix coefficients to m = 0, -1, -2, .. . We discovered that ES1[ -1,n] = 0.5 for n = 1, 2, .. . As usual eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function.
The coefficients in the columns of the ES1 matrix, for m = 1, 2, 3, .. , and n = 2, 3, 4 .. , can be generated with the polynomials GF(z,n) for which we found the following general expression GF(z;n) = ((-1)^(n-1)*r(n)*CFN1(z,n)*GF(z;n=1) + ETA(z,n))/p(n).
The CFN1(z,n) polynomials depend on the central factorial numbers A008955.
The ETA(z,n) are the Eta polynomials which lead to the Eta triangle.
The zero patterns of the Eta polynomials resemble a UFO. These patterns resemble those of the Zeta, Beta and Lambda polynomials, see A160474, A160480 and A160487.
The first Maple algorithm generates the coefficients of the Eta triangle. The second Maple algorithm generates the ES1[2*m-1,n] coefficients for m= 0, -1, -2, -3, .. .
The M(n) sequence, see the second Maple algorithm, leads to Gould's sequence A001316 and a sequence that resembles the denominators of the Taylor series for tan(x), A156769(n).
Some of our results are conjectures based on numerical evidence, see especially A160466.

			The first few rows of the triangle ETA(n,m) with n=2,3,.. and m=1,2,... are
  [ -1]
  [ -11, 2]
  [ -114, 29, -2]
  [ -3963, 1156, -122, 4].
The first few ETA(z,n) polynomials are
  ETA(z,n=2) = -1;
  ETA(z,n=3) = -11+2*z^2;
  ETA(z,n=4) = -114 + 29*z^2 - 2*z^4.
The first few CFN1(z,n) polynomials are
  CFN1(z,n=2) = (z^2-1);
  CFN1(z,n=3) = (z^4 - 5*z^2 + 4);
  CFN1(z,n=4) = (z^6 - 14*z^4 + 49*z^2 - 36).
The first few generating functions GF(z;n) are:
  GF(z;n=2) = ((-1)*2*(z^2 - 1)*GF(z;n=1) + (- 1))/3;
  GF(z;n=3) = (4*(z^4 - 5*z^2+4) *GF(z;n=1) + (-11 + 2*z^2))/30;
  GF(z;n=4) = ((-1)*4*(z^6 - 14*z^4 + 49*z^2 - 36)*GF(z;n=1) + (-114 + 29*z^2 - 2*z^4))/315.

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

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.
J. W. Meijer and N.H.G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.
Eric. W. Weisstein, Dirichlet Eta Function, Wolfram MathWorld.

The r(n) sequence equals A062383 (n>=1).
The p(n) sequence equals A160473(n) (n>=2).
The GCS(n) sequence equals the Geometric Connell sequence A049039(n).
The M(n-1) sequence equals A001316(n-1)/A156769(n) (n>=1).
The q(n) sequence leads to A081729 and the 'gossip sequence' A007456.
The first right hand column equals A053644 (n>=1).
The first left hand column equals A160465.
The row sums equal A160466.
The CFN1(z, n) and the cfn1(n, k) lead to A008955.
Cf. A094665 and A160468.
Cf. the Zeta, Beta and Lambda triangles A160474, A160480 and A160487.
Cf. A162440 (EG1 matrix).

nmax:=8; c(2 ):= -1/3: for n from 3 to nmax do c(n) := (2*n-2)*c(n-1)/(2*n-1)-1/((n-1)*(2*n-1)) end do: for n from 2 to nmax do GCS(n-1) := ln(1/(2^(-(2*(n-1)-1-floor(ln(n-1)/ ln(2))))))/ln(2); p(n) := 2^(-GCS(n-1))*(2*n-1)!; ETA(n, 1) := p(n)*c(n); ETA(n, n) := 0 end do: mmax:=nmax: for m from 2 to mmax do for n from m+1 to nmax do q(n) := (1+(-1)^(n-3)*(floor(ln(n-1)/ln(2)) - floor(ln(n-2)/ln(2)))): ETA(n, m) := q(n)*((-1)*ETA(n-1, m-1)+(n-1)^2*ETA(n-1, m)) end do end do: seq(seq(ETA(n,m), m=1..n-1), n=2..nmax);
# End first program.
nmax1:=20; m:=1; ES1row:=1-2*m; with (combinat): cfn1 := proc(n, k): sum((-1)^j*stirling1(n+1, n+1-k+j) * stirling1(n+1, n+1-k-j), j=-k..k) end proc: mmax1:=nmax1: for m1 from 1 to mmax1 do M(m1-1) := 2^(2*m1-2)/((2*m1-1)!); ES1[-2*m1+1,1] := 2*(1-2^(1-(1-2*m1)))*(-bernoulli(2*m1)/(2*m1)) od: for n from 2 to nmax1 do for m1 from 1 to mmax1-n+1 do ES1[1-2*m1, n] := (-1)^(n-1)*M(n-1)*sum((-1)^(k+1)*cfn1(n-1,k-1)* ES1[2*k-2*n-2*m1+1, 1], k=1..n) od: od: seq(ES1[1-2*m, n], n=1..nmax1-m+1);
# End second program.

We discovered an interesting relation between the Eta triangle coefficients ETA(n,m) = q(n)*((-1)*ETA(n-1,m-1)+(n-1)^2*ETA(n-1,m)), for n = 3, 4, ... and m = 2, 3, ... , with
q(n) = 1 + (-1)^(n-3)*(floor(log(n-1)/log(2)) - floor(log(n-2)/log(2))) for n = 3, 4, ....
See A160465 for ETA(n,m=1) and furthermore ETA(n,n) = 0 for n = 2, 3, ....
The generating functions GF(z;n) of the coefficients in the matrix columns are defined by
GF(z;n) = sum_{m>=1} ES1[2*m-1,n] * z^(2*m-2), with n = 1, 2, 3, .... This leads to
GF(z;n=1) = (2*log(2) - Psi(z) - Psi(-z) + Psi(1/2*z) + Psi(-1/2*z)); Psi(z) is the digamma-function.
GF(z;n) = ((2*n-2)/(2*n-1)-2*z^2/((n-1)*(2*n-1)))*GF(z;n-1)-1/((n-1)*(2*n-1)).
We found for GF(z;n), for n = 2, 3, ..., the following general expression:
GF(z;n) = ((-1)^(n-1)*r(n)*CFN1(z,n)*GF(z;n=1) + ETA(z,n) )/p(n) with
r(n) = 2^floor(log(n-1)/log(2)+1) and
p(n) = 2^(-GCS(n))*(2*n-1)! with
GCS(n) = log(1/(2^(-(2*(n-1)-1-floor(log(n-1)/ log(2))))))/log(2).

A160487 The Lambda triangle.

Original entry on oeis.org

1, -107, 10, 59845, -7497, 210, -6059823, 854396, -35574, 420, 5508149745, -827924889, 41094790, -765534, 4620, -8781562891079, 1373931797082, -75405128227, 1738417252, -17219202, 60060

Offset: 2

Johannes W. Meijer, May 24 2009, Sep 18 2012

The coefficients of the LS1 matrix are defined by LS1[2*m,n] = int(y^(2*m)/(sinh(y))^(2*n-1),y=0..infinity)/factorial(2*m) for m = 1, 2, 3, .. and n = 1, 2, 3, .. under the condition that n <= m.
This definition leads to LS1[2*m,n=1] = 2*lambda(2*m+1), for m = 1, 2, .. , and the recurrence relation LS1[2*m,n] = ((2*n-3)/(2*n-2))*(LS1[2*m-2,n-1]/(2*n-3)^2- LS1[2*m,n-1]). As usual lambda(m) = (1-2^(-m))*zeta(m) with zeta(m) the Riemann zeta function.
These two formulas enable us to determine the values of the LS1[2*m,n] coefficients, for all integers m and all positive integers n, but not for all n. If we choose, somewhat but not entirely arbitrarily, LS1[m=0,n=1] = gamma, with gamma the Euler-Mascheroni constant, we can determine them all.
The coefficients in the columns of the LS1 matrix, for m = 0, 1, 2, .. , and n = 2, 3, 4 .. , can be generated with the GL(z;n) polynomials for which we found the following general expression GL(z;n) = (h(n)*CFN2(z;n)*GL(z;n=1) + LAMBDA(z;n))/p(n).
The CFN2(z;n) polynomials depend on the central factorial numbers A008956.
The LAMBDA(z;n) are the Lambda polynomials which lead to the Lambda triangle.
The zero patterns of the Lambda polynomials resemble a UFO. These patterns resemble those of the Eta, Zeta and Beta polynomials, see A160464, A160474 and A160480.
The first Maple algorithm generates the coefficients of the Lambda triangle. The second Maple algorithm generates the LS1[2*m,n] coefficients for m= -1, -2, -3, .. .
Some of our results are conjectures based on numerical evidence.

			The first few rows of the triangle LAMBDA(n,m) with n=2,3,.. and m=1,2,.. are
  [1]
  [ -107, 10]
  [59845, -7497, 210]
  [ -6059823, 854396, -35574, 420]
The first few LAMBDA(z;n) polynomials are
  LAMBDA (z;n=2) = 1
  LAMBDA (z;n=3) = -107 +10*z^2
  LAMBDA (z;n=4) = 59845-7497*z^2+210*z^4
The first few CFN2(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 GL(z;n) are:
  GL(z;n=2) = (6*(z^2-1)*GL(z,n=1) + (1)) /12
  GL(z;n=3) = (60*(z^4-10*z^2+9)*GL(z,n=1)+ (-107+10*z^2)) / 1440
  GL(z;n=4) = (1260*( z^6- 35*z^4+259*z^2-225)*GL(z,n=1) + (59845-7497*z^2+ 210*z^4))/907200

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

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.

A160488 equals the first left hand column.
A160476 equals the first right hand column and 6*h(n).
A160489 equals the rows sums.
A160490 equals the p(n) sequence.
A160479 equals the ZL(n) sequence.
A001620 is the Euler-Mascheroni constant gamma.
The LS1[ -2, n] coefficients lead to A002197, A002198 and A058962.
The LS1[ -2*m, 1] coefficients equal (-1)^(m+1)*A036282/A036283.
The CFN2(z, n) and the cfn2(n, k) lead to A008956.
Cf. The Eta, Zeta and Beta triangles A160464, A160474 and A160480.
Cf. A162448 (LG1 matrix)

nmax:=7; for n from 0 to nmax do cfn2(n, 0) := 1: cfn2(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax 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: for n from 1 to nmax do Delta(n-1) := sum((1-2^(2*k1-1))* (-1)^(n+1)*(-bernoulli(2*k1)/(2*k1))*(-1)^(k1+n)*cfn2(n-1,n-k1, n), k1=1..n) / (2*4^(n-1)*(2*n-1)!); LAMBDA(-2, n) := sum(2*(1-2^(2*k1-1))*(-bernoulli(2*k1) / (2*k1))*(-1)^(k1+n)* cfn2(n-1,n-k1), k1=1..n)/ factorial(2*n-2) end do: Lcgz(2) := 1/12: f(2) := 1/12: for n from 3 to nmax do Lcgz(n) := LAMBDA(-2, n-1)/((2*n-2)*(2*n-3)): f(n) := Lcgz(n)-((2*n-3)/(2*n-2))*f(n-1) end do: for n from 1 to nmax do b(n) := denom(Lcgz(n+1)) end do: for n from 1 to nmax do b(n) := 2*n*denom(Delta(n-1))/2^(2*n) end do: p(2) := b(1): for n from 2 to nmax do p(n+1) := lcm(p(n)*(2*n)*(2*n-1), b(n)) end do: for n from 2 to nmax do LAMBDA(n, 1) := p(n)*f(n) end do: mmax:=nmax: for n from 2 to nmax do LAMBDA(n, n) := 0 end do: for n from 1 to nmax do b(n) := (2*n)*(2*n-1)*denom(Delta(n-1))/ (2^(2*n)*(2*n-1)) end do: c(1) := b(1): for n from 1 to nmax-1 do c(n+1) := lcm(c(n)*(2*n+2)* (2*n+1), b(n+1)) end do: for n from 1 to nmax do cm(n) := c(n)/(6*(2*n)!) end do: for n from 1 to nmax-1 do ZL(n+2) := cm(n+1)/cm(n) end do: for m from 2 to mmax do for n from m+1 to nmax do LAMBDA(n, m) := ZL(n)*(LAMBDA(n-1, m-1)-(2*n-3)^2*LAMBDA(n-1, m)) end do end do; seq(seq(LAMBDA(n,m), m=1..n-1), n=2..nmax);
# End first program.
nmax1:=10; m:=1; LS1row:=-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 LS1[-2*m1, 1] := 2*(1-2^(-(-2*m1+1)))*(-bernoulli(2*m1)/(2*m1)) od: for n from 2 to nmax1 do for m1 from 1 to mmax1-n+1 do LS1[ -2*m1, n] := sum((-1)^(k1+1)*cfn2(n-1,k1-1)* LS1[2*k1-2*n-2*m1, 1], k1=1..n)/(2*n-2)! od: od: seq(LS1[ -2*m, n], n=1..nmax1-m+1);
# End second program.

We discovered a remarkable relation between the Lambda triangle coefficients Lambda(n,m) = ZL(n)*(Lambda(n-1,m-1)-(2*n-3)^2*Lambda(n-1,m)) for n = 3, 4, .. and m = 2, 3, .. . See A160488 for LAMBDA(n,m=1) and furthermore LAMBDA(n,n) = 0 for n = 2, 3, .. .
We observe that the ZL(n) = A160479(n) sequence also rules the Zeta triangle A160474.
The generating functions GL(z;n) of the coefficients in the matrix columns are defined by
GL(z;n) = sum(LS1[2*m-2,n]*z^(2*m-2), m=1..infinity), with n = 1, 2, 3, .. .
This definition, and our choice of LS1[m=0,n=1] = gamma, leads to GL(z;n=1) = -2*Psi(1-z)+Psi(1-(z/2))-(Pi/2)*tan(Pi*z/2) with Psi(z) the digamma-function. Furthermore we discovered that GL(z;n) =GL(z;n-1)*(z^2/((2*n-2)*(2*n-3)) -(2*n-3)/((2*n-2)))+LS1[ -2,n-1]/((2*n-2)*(2*n-3)) for n = 2, 3 , .. . with LS1[ -2,n] = (-1)^(n-1)*4*A058962(n-1)*A002197(n-1)/A002198(n-1) for n = 1, 2, .. , with A058962(n-1) = 2^(2*n-2)*(2*n-1).
We found the following general expression for the GL(z;n) polynomials, for n = 2, 3, ..
GL(z;n) = (h(n)*CFN2(z;n)*GL(z;n=1) + LAMBDA(z;n))/p(n) with
h(n) = 6*A160476(n) and p(n) = A160490(n).

A160474 The Zeta triangle.

Original entry on oeis.org

-1, 51, -10, -10594, 2961, -210, 356487, -115940, 12642, -420, -101141295, 35804857, -4751890, 254562, -4620, 48350824787, -18071509911, 2689347661, -180909586, 5471466, -60060

Offset: 2

Johannes W. Meijer, May 24 2009

The coefficients of the ZS1 matrix are defined by ZS1[2*m-1,n] = (2^(2*m-1))*int(y^(2*m-1)/(sinh(y))^(2*n), y=0..infinity)/factorial(2*m-1) for m = 1, 2, 3, .. and n = 1, 2, 3, .. under the condition that n <= (m-1).
This definition leads to ZS1[2*m-1,n=1] = 2*zeta(2*m-1), for m = 2, 3, .. , and the recurrence relation ZS1[2*m-1,n]:=(1/(2*n-1))*((2/(n-1))*ZS1[2*m-3,n-1]-(2*n-2)*ZS1[2*m-1,n-1]). As usual zeta(m) is the Riemann zeta function. These two formulas enable us to determine the values of the ZS[2*m-1,n] coefficients, with m all integers and n all positive integers, but not for all. If we choose, somewhat but not entirely arbitrarily, ZS1[1,n=1] = 2*gamma, with gamma the Euler-Mascheroni constant, we can determine them all.
The coefficients in the columns of the ZS1 matrix, for m = 1, 2, 3, .., and n = 2, 3, 4 .. , can be generated with the GH(z;n) polynomials for which we found the following general expression GH(z;n) = (h(n)*CFN1(z;n)*GH(z;n=1) + ZETA(z;n))/p(n).
The CFN1(z;n) polynomials depend on the central factorial numbers A008955.
The ZETA(z;n) are the Zeta polynomials which lead to the Zeta triangle.
The zero patterns of the Zeta polynomials resemble a UFO. These patterns resemble those of the Eta, Beta and Lambda polynomials, see A160464, A160480 and A160487.
The first Maple algorithm generates the coefficients of the Zeta triangle. The second Maple algorithm generates the ZS1[2*m-1,n] coefficients for m= 0, -1, -2, .. .
The M(n) sequence, see the second Maple algorithm, leads to Gould's sequence A001316 and a sequence that resembles the denominators in Taylor series for tan(x), i.e., A156769(n).
Some of our results are conjectures based on numerical evidence.

			The first few rows of the triangle ZETA(n,m) with n=2,3,... and m=1,2,... are
  [ -1],
  [51, -10],
  [ -10594, 2961, -210],
  [356487, -115940, 12642, -420].
The first few ZETA(z;n) polynomials are
  ZETA(z;n=2) = -1,
  ZETA(z;n=3) = 51-10*z^2,
  ZETA(z;n=4) = -10594 + 2961*z^2 - 210*z^4.
The first few CFN1(z;n) polynomials are
  CFN1(z;n=2) = (z^2-1),
  CFN1(z;n=3) = (z^4 - 5*z^2 + 4),
  CFN1(z;n=4) = (z^6 - 14*z^4 + 49*z^2 - 36).
The first few generating functions GH(z;n) are
  GH(z;n=2) = (6*(z^2-1)*GH(z;n=1) + (-1)) / 9,
  GH(z;n=3) = (60*(z^4-5*z^2+4)*GH(z;n=1) + (51-10*z^2)) / 450,
  GH(z;n=4) = (1260*(z^6-14*z^4+49*z^2-36)*GH(z;n=1) + (-10594+2961*z^2-210*z^4))/99225.

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.
Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
Johannes W. Meijer, The zeros of the Eta, Zeta, Beta and Lambda polynomials, jpg and pdf, Mar 03 2013.
Johannes W. Meijer and N.H.G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.

A160475 equals the first left hand column.
A160476 equals the first right hand column and 6*h(n).
A160477 equals the rows sums.
A160478 equals the p(n) sequence.
A160479 equals the ZL(n) sequence.
A001620 is the Euler-Mascheroni constant gamma.
The M(n-1) sequence equals A001316(n-1)/A156769(n) (n>=1).
The ZS1[ -1, n] and the Omega(n) coefficients lead to A002195 and A002196.
The CFN1(z, n) and the cfn1(n, k) lead to A008955.
Cf. The Eta, Beta and Lambda triangles A160464, A160480 and A160487.
Cf. A162446 (ZG1 matrix)

nmax:=7; with(combinat): cfn1 := proc(n, k): sum((-1)^j*stirling1(n+1, n+1-k+j) * stirling1(n+1, n+1-k-j), j = -k..k) end proc: Omega(0):=1: for n from 1 to nmax do Omega(n) := (sum((-1)^(k1+n+1)*(bernoulli(2*k1)/(2*k1))*cfn1(n-1, n-k1), k1=1..n))/(2*n-1)! end do: for n from 1 to nmax do Zc(n) := (Omega(n)*2^(2*n-1))*2/((2*n+1)*(n)) end do: c(1) := denom(Zc(1)): for n from 2 to nmax do c(n) := lcm(c(n-1)*(n)*(2*n+1)/2, denom(Zc(n))); p(n) := c(n-1) end do: y(1):=Zc(1): for n from 1 to nmax-1 do y(n+1) := Zc(n+1) - ((2*n+2)/(2*n+3))*y(n) end do: for n from 1 to nmax do b(n) := 4^(-n)*(2*n+1)*n*denom(Omega(n)) end do: for n from 1 to nmax-1 do c(n+1) := lcm(c(n)*(n+1)*(2*n+3)/2, b(n+1)) end do: for n from 1 to nmax do cm(n) := c(n)*(1/6)* 4^n/(2*n+1)! end do: for n from 1 to nmax-1 do ZL(n+2) := cm(n+1)/cm(n) end do: mmax := nmax: for n from 2 to nmax do ZETA(n, 1) := p(n)*y(n-1): ZETA(n, n) := 0 end do: for m from 2 to mmax do for n from m+1 to nmax do ZETA(n, m) := ZL(n)*(ZETA(n-1, m-1) - (n-1)^2* ZETA(n-1, m)) end do end do; seq(seq(ZETA(n,m), m=1..n-1), n=2..nmax);
# End first program (program edited, Johannes W. Meijer, Sep 20 2012)
nmax1 := 10; m := 1; ZS1row := 1-2*m; with(combinat): t1 := proc(n, k): sum((-1)^j * stirling1(n+1, n+1-k+j) * stirling1(n+1, n+1-k-j), j = -k..k) end proc: mmax1 := nmax1: for m1 from 1 to mmax1 do M(m1-1) := 2^(2*m1-2)/((2*m1-1)!) end do: for m1 from 1 to mmax1 do ZS1[ -2*m1+1, 1] := 2*(-bernoulli(2*m1)/(2*m1)) od: for n from 2 to nmax1 do for m1 from 1 to mmax1-n+1 do ZS1[-2*m1+1, n] := M(n-1)*sum((-1)^(k1+1)*t1(n-1, k1-1) * ZS1[2*k1-2*n-2*m1+1, 1], k1 = 1..n) od: od: seq(ZS1[1-2*m, n], n = 1..nmax1-m+1);
# End second program (program edited, Johannes W. Meijer, Sep 20 2012)

We discovered a remarkable relation between the Zeta triangle coefficients ZETA(n,m) = ZL(n)*(ZETA(n-1,m-1)-(n-1)^2*ZETA(n-1,m)) for n = 3, 4, ... and m = 2, 3, .... See A160475 for ZETA(n,m=1) and furthermore ZETA(n,n) = 0 for n = 2, 3, ....
We observe that the ZL(n) = A160479(n) sequence also rules the Lambda triangle A160487.
The generating functions GH(z;n) of the coefficients in the matrix columns are defined by
GH(z;n) = sum(ZS1[2*m-1,n]*z^(2*m-2), m=1..infinity), with n = 1, 2, 3, .... This definition, and our choice of ZS1[1,1] = 2*gamma, leads to GH(z;n=1) = (-Psi(1-z)-Psi(1+z)) with Psi(z) the digamma-function. Furthermore we discovered that GH(z;n) = GH(z;n-1)*(2*z^2/((2*n-1)*(n-1))-(2*n-2)/(2*n-1))+2*ZS1[ -1,n-1]/((2*n-1)*(n-1)) for n = 2, 3 , ..., with ZS1[ -1,n] = 2^(2*n-1)*A002195(n)/A002196(n) for n = 1, 2, ....
We found the following general expression for the GH(z;n) polynomials, for n = 2, 3, ...:
GH(z;n) = (h(n)*CFN1(z;n)*GH(z;n=1) + ZETA(z;n))/p(n) with
h(n) = 6*A160476(n) and p(n) = A160478(n).

A160485 Triangle of the RBS1 polynomial coefficients.

Original entry on oeis.org

1, 1, -2, 1, -8, 12, 1, -2, 60, -120, 1, -128, -168, 0, 1680, 1, 2638, 7320, -5040, -25200, -30240, 1, -98408, -300828, 52800, 1053360, 1330560, 665280, 1, 5307118, 17914260, 2522520, -56456400, -90810720, -60540480, -17297280

Offset: 1

Johannes W. Meijer, May 24 2009, Jul 06 2009, Sep 19 2012

In A160480 we defined the BS1 matrix by BS1[2*m-1,n=1] = 2*beta(2*m) 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), for positive and negative values of m and n= 1, 2, .. . As usual beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). It is well-known that BS1[1-2*m,n=1] = euler(2*m-2) for m = 1, 2, .., with euler(2*m-2) the Euler numbers A000364. These values together with the recurrence relation lead to BS1[ -1,n] = 1 for n = 1, 2, .. .
We discovered that the n-th term of the row coefficients BS1[1-2*m,n] for m = 1, 2, .., can be generated with the rather simple polynomials RBS1(1-2*m,n). Our discovery was enabled by the recurrence relation for the RBS1(1-2*m,n) polynomials which we derived from the recurrence relation for the BS[2*m-1,n] coefficients and the fact that RBS1(-1,n) = 1.
The RBS1 polynomials and the polynomials defined by sequence A083061 are related by a shift of +-1/2 and scaling by a power of 2 (see arXiv link). - Richard P. Brent, Jul 15 2014

			The first few rows of the triangle are:
[1]
[1, -2]
[1, -8, 12]
[1, -2, 60, -120]
[1, -128, -168, 0, 1680]
The first few RBS1(1-2*m,n) polynomials are:
RBS1(-1,n) = 1
RBS1(-3,n) = 1 - 2*n
RBS1(-5,n) = 1 - 8*n + 12*n^2
RBS1(-7,n) = 1 - 2*n + 60*n^2 - 120*n^3
From _Peter Bala_, Jan 22 2019: (Start)
Qbar(r,n) = binomial(2*n+2,n+1)/(2^(2*n+1)) * Sum_{k = 0..n} binomial(n,k)/binomial(n+k+1,k)*(2*k + 1)^(2*r):
Case r = 2: Qbar(2,n) = binomial(2*n+2,n+1)/2^(2*n+1) * ( 1 + 3^4*n/(n+2) + 5^4*n*(n-1)/((n+2)*(n+3)) + 7^4*n*(n-1)*(n-2)/((n+2)*(n+3)*(n+4)) + ... ) = 12*n^2 + 8*n + 1, valid for n a nonnegative integer (when the series terminates). The identity is also valid for complex n with real part greater than 1 (provided the factor binomial(2*n,n) is replaced with the appropriate expression involving the gamma function).
Case r = 3: Qbar(3,n) = binomial(2*n+2,n+1)/(2^(2*n+1)) * ( 1 + 3^6*n/(n+2) + 5^6*n*(n-1)/((n+2)*(n+3)) + 7^6*n*(n-1)*(n-2)/((n+2)*(n+3)*(n+4)) + ... ) = 120*n^3 + 60*n^2 + 2*n + 1, valid for n a nonnegative integer. The identity is also valid for complex n with real part greater than 2.
Note, the case r = 0 is equivalent to the identity 1 = binomial(2*n,n)/2^(2*n-1) * ( 1 + (n-1)/(n+1) + (n-1)*(n-2)/((n+1)*(n+2)) + (n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)) + ... ), which is valid for complex n with real part greater than 0. This identity was found by Ramanujan. See Example 6, Chapter 10 in Berndt. (End)

B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, Chapter 10, p. 21.

P. Bala, A245244 and A160485 and some hypergeometric series evaluations of Ramanujan
R. P. Brent, Generalising Tuenter's binomial sums, arXiv:1407.3533 [math.CO], 2014.
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.

A160480 is the Beta triangle.
A009389(2*n) equals the second left hand column divided by 2.
A001813 equals the first right hand column.
The absolute values of the row sums equal the Euler numbers A000364.
Cf. A083061, A036970, A245244.

nmax := 8; mmax := nmax: A(1, 1) := 1: RBS1(n, 2) := (2*n-1)^2*1-(2*n)*(2*n-1)*1: for m from 3 to mmax do for k from 0 to m-1 do A(m-1, k+1) := coeff(RBS1(n, m-1), n, k) od; RBS1(n+1, m-1) := 0: for k from 0 to m-1 do RBS1(n+1, m-1) := RBS1(n+1, m-1) + A(m-1, k+1)*(n+1)^k od: RBS1(n, m) := (2*n-1)^2*RBS1(n, m-1)-(2*n)*(2*n-1) * RBS1(n+1, m-1) od: for k from 0 to nmax-1 do A(nmax, k+1) := coeff(RBS1(n, nmax), n, k) od: seq(seq(A(n, m), m=1..n), n=1..nmax);

RBS1(1-2*m,n) = (2*n-1)^2*RBS1(3-2*m,n)-(2*n)*(2*n-1)*RBS1(3-2*m,n+1) for m = 2, 3, .., with RBS1(-1,n) =1 for n = 1, 2, .. .
From Peter Bala, Jan 22 2019: (Start)
The row polynomials RBS1 of the triangle are related to the polynomials Qbar(r,n), r = 0,1,2,..., introduced by Brent by Qbar(r,n) = RBS1(-2*r-1,-n).
Recurrence: Qbar(r+1,n) = (2*n + 1)^2*Qbar(r,n) - 2*n(2*n + 1)*Qbar(r,n-1) with Qbar(0,n) = 1 (Brent, equation 19, p.7).
Qbar(r,n) = binomial(2*n + 2,n + 1)/(2^(2*n + 1)) * Sum_{k = 0..n} binomial(n,k)/binomial(n + k + 1,k)*(2*k + 1)^(2*r); this follows easily from the above recurrence. Two examples are given below.
Qbar(r,n) = 1/4^n * Sum_{k = 0..n} binomial(2*n + 1,n - k)*(2*k + 1)^(2*r) For related polynomial sequences see A036970, A083061 and A245244. (End)

A162443 Numerators of the BG1[ -5,n] coefficients of the BG1 matrix.

Original entry on oeis.org

5, 66, 680, 2576, 33408, 14080, 545792, 481280, 29523968, 73465856, 27525120, 856162304, 1153433600, 18798870528, 86603988992, 2080374784, 2385854332928, 3216930504704, 71829033058304, 7593502179328, 281749854617600

Offset: 1

Johannes W. Meijer, Jul 06 2009

The BG1 matrix coefficients are defined by BG1[2m-1,1] = 2*beta(2m) and the recurrence relation BG1[2m-1,n] = BG1[2m-1,n-1] - BG1[2m-3,n-1]/(2*n-3)^2 with m = .. , -2, -1, 0, 1, 2, .. and n = 1, 2, 3, .. . As usual beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). For the BG2 matrix, the even counterpart of the BG1 matrix, see A008956.
We discovered that the n-th term of the row coefficients can be generated with BG1[1-2*m,n] = RBS1(1-2*m,n)* 4^(n-1)*((n-1)!)^2/ (2*n-2)! for m >= 1. For the BS1(1-2*m,n) polynomials see A160485.
The coefficients in the columns of the BG1 matrix, for m >= 1 and n >= 2, can be generated with GFB(z;n) = ((-1)^(n+1)*CFN2(z;n)*GFB(z;n=1) + BETA(z;n))/((2*n-3)!!)^2 for n >= 2. For the CFN2(z;n) and the Beta polynomials see A160480.
The BG1[ -5,n] sequence can be generated with the first Maple program and the BG1[2*m-1,n] matrix coefficients can be generated with the second Maple program.
The BG1 matrix is related to the BS1 matrix, see A160480 and the formulas below.

			The first few formulas for the BG1[1-2*m,n] matrix coefficients are:
BG1[ -1,n] = (1)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -3,n] = (1-2*n)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!
The first few generating functions GFB(z;n) are:
GFB(z;2) = ((-1)*(z^2-1)*GFB(z;1) + (-1))/1
GFB(z;3) = ((+1)*(z^4-10*z^2+9)*GFB(z;1) + (-11 + z^2))/9
GFB(z;4) = ((-1)*( z^6- 35*z^4+259*z^2-225)*GFB(z;1) + (-299 + 36*z^2 - z^4))/225

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.

A162444 are the denominators of the BG1[ -5, n] matrix coefficients.
The BG1[ -3, n] equal (-1)*A002595(n-1)/A055786(n-1) for n >= 1.
The BG1[ -1, n] equal A046161(n-1)/A001790(n-1) for n >= 1.
The cs(n) equal A046161(n-2)/A001803(n-2) for n >= 2.
The BETA(z, n) polynomials and the BS1 matrix lead to the Beta triangle A160480.
The CFN2(z, n), the t2(n, m) and the BG2 matrix lead to A008956.
Cf. A000364, A001818 and A160485.
Cf. A162443 (BG1 matrix), A162446 (ZG1 matrix) and A162448 (LG1 matrix).

a := proc(n): numer((1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!) end proc: seq(a(n), n=1..21);
# End program 1
nmax1 := 5; coln := 3; Digits := 20: mmax1 := nmax1: for n from 0 to nmax1 do t2(n, 0) := 1 od: for n from 0 to nmax1 do t2(n, n) := doublefactorial(2*n-1)^2 od: for n from 1 to nmax1 do for m from 1 to n-1 do t2(n, m) := (2*n-1)^2* t2(n-1, m-1) + t2(n-1, m) od: od: for m from 1 to mmax1 do BG1[1-2*m, 1] := euler(2*m-2) od: for m from 1 to mmax1 do BG1[2*m-1, 1] := Re(evalf(2*sum((-1)^k1/(1+2*k1)^(2*m), k1=0..infinity))) od: for m from -mmax1 +coln to mmax1 do BG1[2*m-1, coln] := (-1)^(coln+1)*sum((-1)^k1*t2(coln-1, k1)*BG1[2*m-(2*coln-1)+2*k1, 1], k1=0..coln-1)/doublefactorial(2*coln-3)^2 od;
# End program 2
# Maple programs edited by Johannes W. Meijer, Sep 25 2012

a(n) = numer(BG1[ -5,n]) and A162444(n) = denom(BG1[ -5,n]) with BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!.
The generating functions GFB(z;n) of the coefficients in the matrix columns are defined by
GFB(z;n) = sum(BG1[2*m-1,n]*z^(2*m-2), m=1..infinity).
GFB(z;n) = (1-z^2/(2*n-3)^2)*GFB(n-1) - 4^(n-2)*(n-2)!^2/((2*n-4)!*(2*n-3)^2) for n => 2 with GFB(z;n=1) = 1/(z*cos(Pi*z/2))*int(sin(z*t)/sin(t),t=0..Pi/2).
The column sums cs(n) = sum(BG1[2*m-1,n]*z^(2*m-2), m=1..infinity) = 4^(n-1)/((2*n-2)*binomial(2*n-2,n-1)) for n >= 2.
BG1[2*m-1,n] = (n-1)!^2*4^(n-1)*BS1[2*m-1,n]/(2*n-2)!

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A160481 Row sums of the Beta triangle A160480.

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A160482 Second left hand column of the Beta triangle A160480.

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A160483 Second right hand column of the Beta triangle A160480.

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A160484 Third right hand column of the Beta triangle A160480.

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A008956 Triangle of central factorial numbers |4^k t(2n+1,2n+1-2k)| read by rows (n>=0, k=0..n).

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A160464 The Eta triangle.

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A160487 The Lambda triangle.

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A160474 The Zeta triangle.

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