A160474 -id:A160474 - cp's OEIS frontend

A160479 The ZL(n) sequence of the Zeta and Lambda triangles A160474 and A160487.

10, 21, 2, 11, 13, 1, 34, 57, 5, 23, 1, 1, 29, 31, 2, 1, 37, 1, 41, 301, 1, 47, 1, 1, 53, 3, 1, 59, 61, 1, 2, 67, 1, 71, 73, 1, 1, 79, 1, 83, 1, 1, 89, 1, 1, 1, 97, 1, 505, 103, 1, 107, 109, 11, 113, 1, 1, 1, 1, 1, 1, 127, 2, 131

Offset: 3

Johannes W. Meijer, May 24 2009

The rather strange ZL(n) sequence rules both the Zeta and Lambda triangles.
The Zeta triangle led to the first and the Lambda triangle to the second Maple algorithm.
The first ZL(n) formula is a conjecture. This formula links the ZL(n) to the prime numbers A000040; see A217983, A128060, A130290 and the third Maple program.

Cf. A160474 and A160487.
The cnf1(n, k) are the central factorial numbers A008955.
The cnf2(n, k) are the central factorial numbers A008956.

nmax := 65; for n from 0 to nmax do cfn1(n, 0):=1: cfn1(n, n):=(n!)^2 end do: for n from 1 to nmax do for k from 1 to n-1 do cfn1(n, k) := cfn1(n-1, k-1)*n^2 + cfn1(n-1, k) end do: end do: 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 d(n) := 2^(2*n-1)*Omega(n) end do: for n from 1 to nmax do b(n) := 4^(-n)*(2*n+1)*n*denom(Omega(n)) end do: c(1) := b(1): 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 3 to nmax+1 do ZL(n):=cm(n-1)/cm(n-2) end do: seq(ZL(n), n=3..nmax+1);
# End program 1 (program edited by Johannes W. Meijer, Oct 25 2012)
nmax1 := nmax; 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: for n from 1 to nmax1 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), k1=1..n) /(2*4^(n-1)*(2*n-1)!) end do: for n from 1 to nmax1 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 nmax1-1 do c(n+1) := lcm(c(n)*(2*n+2)* (2*n+1), b(n+1)) end do: for n from 1 to nmax1 do cm(n) := c(n)/(6*(2*n)!) end do: for n from 3 to nmax1+1 do ZL(n) := cm(n-1)/cm(n-2) end do: seq(ZL(n), n=3..nmax1+1);
# End program 2 (program edited by Johannes W. Meijer, Sep 20 2012)
nmax2 := nmax: A000040 := proc(n): ithprime(n) end: A130290 := proc(n): if n =1 then 1 else (A000040(n)-1)/2 fi: end: A128060 := proc(n) local n1: n1:=2*n-1: if type(n1, prime) then A128060(n) := 1 else A128060(n) := n1 fi: end: for n from 1 to nmax2 do A217983(n) := 1 od: for n from 1 to nmax2 do for n1 from 1 to floor(log[A000040(n)](nmax2)) do A217983(A130290(n) * A000040(n)^n1) := A000040(n) od: od: ZL := proc(n): (2*n-1)*(A217983(n-1)/A128060(n)) end: seq(ZL(n), n=3..nmax2+1);
# End program 3 (program added by Johannes W. Meijer, Oct 25 2012)

ZL(n) = (2*n-1) * (A217983(n-1)/A128060(n)) for n >= 3.
ZL(n) = ZETA(n, m)/(ZETA(n-1, m-1) - (n-1)^2 * ZETA(n-1, m)), see A160474.
ZL(n) = LAMBDA(n, m)/(LAMBDA(n-1, m-1) - (2*n-3)^2 * LAMBDA(n-1, m)), see A160487.
ZL(n) = A160476(n)/A160476(n-1).

Comments, formulas and third Maple program added by Johannes W. Meijer, Oct 25 2012

A160478 The p(n) sequence that is associated with the Zeta triangle A160474.

Original entry on oeis.org

9, 450, 99225, 3572100, 1080560250, 547844046750, 28761812454375, 66497310394515000, 324074642207668852500, 170139187159026147562500, 495019965039186576333093750, 74252994755877986449964062500

Offset: 2

Johannes W. Meijer, May 24 2009

Cf. A160474 and A160476.

nmax:=15: 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 d(n) := 2^(2*n-1)*Omega(n) end do: for n from 2 to nmax do Zc(n-1) := d(n-1)*2/((2*n-1)*(n-1)) end do: c(1) := denom(Zc(1)): for n from 1 to nmax-1 do c(n+1) := lcm(c(n)*(n+1)*(2*n+3)/2, denom(Zc(n+1))): p(n+1) := c(n) end do: seq(p(n), n=2..nmax);
#  (program edited, Johannes W. Meijer, Sep 20 2012)

a(n) = 3*2^(3-2*n)*(2*n-1)!*A160476(n), for n = 2, 3, .. , with A160476 the first right hand column of the Zeta triangle.

A160475 First left hand column of the Zeta triangle A160474.

Original entry on oeis.org

-1, 51, -10594, 356487, -101141295, 48350824787, -2405967772180, 5296878246375849, -24680641353374049205, 12431632076904547636178, -34807634670487142385955264, 5037797143580320963623681605

Offset: 2

Johannes W. Meijer, May 24 2009

A160474 is the Zeta triangle.

nmax:=13; 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 d(n):= 2^(2*n-1)*Omega(n) end do: for n from 2 to nmax do Zc(n-1) := d(n-1)*2/((2*n-1)*(n-1)) end do: c(1) := denom(Zc(1)): for n from 1 to nmax-1 do c(n+1) := lcm(c(n)*(n+1)*(2*n+3)/2, denom(Zc(n+1))); p(n+1) := c(n) end do: y(1) := Zc(1): for n from 1 to nmax-2 do y(n+1) := Zc(n+1)-((2*n+2)/(2*n+3))*y(n) end do: for n from 2 to nmax do ZETA(n, 1) := p(n)*y(n-1) end do: seq(ZETA(n, 1), n=2..nmax);
# edited, Johannes W. Meijer, Sep 20 2012

A160477 Row sums of the Zeta triangle A160474.

Original entry on oeis.org

-1, 41, -7843, 252769, -69838386, 32793164357, -1610849246753, 3511931460657095, -16239361950453107572, 8129795565691811281818, -22648638361771185452289570, 3264253036763451979723532099

Offset: 2

Johannes W. Meijer, May 24 2009

Cf. A160474 (the Zeta triangle).

# needs Maple code for Zeta triangle from A160474
for n from 2 to nmax do s(n):=0: for m from 1 to n-1 do s(n):=s(n) + ZETA(n,m) od: od: seq(s(n), n=2..nmax);

A008955 Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 4, 1, 14, 49, 36, 1, 30, 273, 820, 576, 1, 55, 1023, 7645, 21076, 14400, 1, 91, 3003, 44473, 296296, 773136, 518400, 1, 140, 7462, 191620, 2475473, 15291640, 38402064, 25401600, 1, 204, 16422, 669188, 14739153, 173721912, 1017067024, 2483133696, 1625702400

Offset: 0

N. J. A. Sloane

Discussion of Central Factorial Numbers by N. J. A. Sloane, Feb 01 2011: (Start)
Here is Riordan's definition of the central factorial numbers t(n,k) given in Combinatorial Identities, Section 6.5:
For n >= 0, expand the polynomial
x^[n] = x*Product{i=1..n-1} (x+n/2-i) = Sum_{k=0..n} t(n,k)*x^k.
The t(n,k) are not always integers. The cases n even and n odd are best handled separately.
For n=2m, we have:
x^[2m] = Product_{i=0..m-1} (x^2-i^2) = Sum_{k=1..m} t(2m,2k)*x^(2k).
E.g. x^[8] = x^2(x^2-1^2)(x^2-2^2)(x^2-3^2) = x^8-14x^6+49x^4-36x^2,
which corresponds to row 4 of the present triangle.
So the m-th row of the present triangle gives the absolute values of the coefficients in the expansion of Product_{i=0..m-1} (x^2-i^2).
Equivalently, and simpler, the n-th row gives the coefficients in the expansion of Product_{i=1..n-1}(x+i^2), highest powers first.
For n odd, n=2m+1, we have:
x^[2m+1] = x*Product_{i=0..m-1}(x^2-((2i+1)/2)^2) = Sum_{k=0..m} t(2m+1,2k+1)*x^(2k+1).
E.g. x^[5] = x(x^2-(1/2)^2)(x^2-(3/2)^2) = x^5-10x^3/4+9x/16,
which corresponds to row 2 of the triangle in A008956.
We now rescale to get integers by replacing x by x/2 and multiplying by 2^(2m+1) (getting 1, -10, 9 from the example).
The result is that row m of triangle A008956 gives the coefficients in the expansion of x*Product_{i=0..m} (x^2-(2i+1)^2).
Equivalently, and simpler, the n-th row of A008956 gives the coefficients in the expansion of Product_{i=0..n-1} (x+(2i+1)^2), highest powers first.
Note that the n-th row of A182867 gives the coefficients in the expansion of Product_{i=1..n} (x+(2i)^2), highest powers first.
(End)
Contribution from Johannes W. Meijer, Jun 18 2009: (Start)
We define Beta(n-z,n+z)/Beta(n,n) = Gamma(n-z)*Gamma(n+z)/Gamma(n)^2 = sum(EG2[2m,n]*z^(2m), m = 0..infinity) with Beta(z,w) the Beta function. The EG2[2m,n] coefficients are quite interesting, see A161739. Our definition leads to EG2[2m,1] = 2*eta(2m) and the recurrence relation EG2[2m,n] = EG2[2m,n-1] - EG2[2m-2,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. We found for the matrix coefficients EG2[2m,n] = sum((-1)^(k+n)*t1(n-1,k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2,k=1..n) with the central factorial numbers t1(n,m) as defined above, see also the Maple program.
From the EG2 matrix we arrive at the ZG2 matrix, see A161739 for its odd counterpart, which is defined by ZG2[2m,1] = 2*zeta(2m) and the recurrence relation ZG2[2m,n] = ZG2[2m-2,n-1]/(n*(n-1))-(n-1)*ZG2[2m,n-1]/n for m = -2, -1, 0, 1, 2, ... and n = 2, 3, ... . We found for the ZG2[2m,n] = Sum_{k=1..n} (-1)^(k+1)*t1(n-1,k-1)* 2* zeta(2*m-2*n+2*k)/((n-1)!*(n)!), and we see that the central factorial numbers t1(n,m) once again play a crucial role.
(End)

			Triangle begins:
  1;
  1,   1;
  1,   5,   4;
  1,  14,  49,  36;
  1,  30, 273, 820, 576;
  ...

B. C. Berndt, Ramanujan's Notebooks Part 1, Springer-Verlag 1985.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

Alois P. Heinz, Rows n = 0..100, flattened (first 51 rows from T. D. Noe)
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.
R. H. Boels, Three particle superstring amplitudes with massive legs, arXiv preprint arXiv:1201.2655 [hep-th], 2012.
R. H. Boels and T. Hansen, String theory in target space, arXiv preprint arXiv:1402.6356 [hep-th], 2014.
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).
M. W. Coffey and M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.
T. L. Curtright and T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arxiv:1408.0767 [math-ph], 2014.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
Toshiki Matsusaka, Applications of Faà di Bruno's formula to partition traces, arXiv:2507.00404 [math.NT], 2025. See p. 5.
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.
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
S. Shadrin, L. Spitz, and D. Zvonkine, On double Hurwitz numbers with completed cycles, J. Lond. Math. Soc., II. Ser. 86, No. 2, 407-432 (2012), Corollary 7.5.

Cf. A036969.
Columns include A000330, A000596, A000597. Right-hand columns include A001044, A001819, A001820, A001821. Row sums are in A101686.
Appears in A160464 (Eta triangle), A160474 (Zeta triangle), A160479 (ZL(n)), A161739 (RSEG2 triangle), A161742, A161743, A002195, A002196, A162440 (EG1 matrix), A162446 (ZG1 matrix) and A163927. - Johannes W. Meijer, Jun 18 2009, Jul 06 2009 and Aug 17 2009
Cf. A234324 (central terms).

T:= function(n,k)
    if k=0 then return 1;
    elif k=n then return (Factorial(n))^2;
    else return n^2*T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..8], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Sep 14 2019

a008955 n k = a008955_tabl !! n !! k
a008955_row n = a008955_tabl !! n
a008955_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 + 1
-- Reinhard Zumkeller, Dec 24 2013

T:= func< n,k | Factorial(2*(n+1))*(&+[(-1)^j*Binomial(n,k-j)*(&+[2^(m-2*k)*StirlingFirst(2*(n-k+1)+m, 2*(n-k+1))*Binomial(2*(n-k+1)+2*j-1, 2*(n-k+1)+m-1)/Factorial(2*(n-k+1)+m): m in [0..2*j]]): j in [0..k]]) >;
[T(n,k): k in [0..n], n in [0..8]]; // G. C. Greubel, Sep 14 2019

nmax:=7: for n from 0 to nmax do t1(n, 0):=1: t1(n, n):=(n!)^2 end do: for n from 1 to nmax do for k from 1 to n-1 do t1(n, k) := t1(n-1, k-1)*n^2 + t1(n-1, k) end do: end do: seq(seq(t1(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jun 18 2009, Revised Sep 16 2012
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: # Mircea Merca, Apr 02 2012
# third Maple program:
T:= proc(n, k) option remember; `if`(k=0, 1,
      add(T(j-1, k-1)*j^2, j=1..n))
    end:
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Feb 19 2022

t[n_, 0]=1; t[n_, n_]=(n!)^2; t[n_ , k_ ]:=t[n, k] = n^2*t[n-1, k-1] + t[n-1, k]; Flatten[Table[t[n, k], {n,0,8}, {k,0,n}] ][[1 ;; 42]]
(* Jean-François Alcover, May 30 2011, after recurrence formula *)

T(n,m):=(2*(n+1))!*sum((-1)^k*binomial(n,m-k)*sum((2^(i-2*m)*stirling1(2*(n-m+1)+i,2*(n-m+1))*binomial(2*(n-m+1)+2*k-1,2*(n-m+1)+i-1))/(2*(n-m+1)+i)!,i,0,2*k),k,0,m); /* Vladimir Kruchinin, Oct 05 2013 */

T(n,k)=if(k==0,1, if(k==n, (n!)^2, n^2*T(n-1, k-1) + T(n-1, k)));
for(n=0,8, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Sep 14 2019

# This triangle is (0,0)-based.
def A008955(n, k) :
    if k==0 : return 1
    if k==n : return factorial(n)^2
    return n^2*A008955(n-1, k-1) + A008955(n-1, k)
for n in (0..7) : print([A008955(n, k) for k in (0..n)]) # Peter Luschny, Feb 04 2012

The n-th row gives the coefficients in the expansion of Product_{i=1..n-1}(x+i^2), highest powers first (see Comments section).
The triangle can be obtained from the recurrence t1(n,k) = n^2*t1(n-1,k-1) + t1(n-1,k) with t1(n,0) = 1 and t1(n,n) = (n!)^2.
t1(n,k) = Sum_{j=-k..k} (-1)^j*s(n+1,n+1-k+j)*s(n+1,n+1-k-j) = Sum_{j=0..2*(n+1-k)} (-1)^(n+1-k+j)*s(n+1,j)*s(n+1,2*(n+1-k)-j), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 02 2012
E.g.f.: cosh(2/sqrt(t)*asin(sqrt(t)*z/2)) = 1 + z^2/2! + (1 + t)*z^4/4! + (1 + 5*t + 4*t^2)*z^6/6! + ... (see Berndt, p.263 and p.306). - Peter Bala, Aug 29 2012
T(n,m) = (2*(n+1))!*Sum_{k=0..m} ((-1)^k*binomial(n,m-k)*Sum_{i=0..2*k} ((2^(i-2*m)*stirling1(2*(n-m+1)+i,2*(n-m+1))*binomial(2*(n-m+1)+2*k-1, 2*(n-m+1)+i-1))/(2*(n-m+1)+i)!)). - Vladimir Kruchinin, Oct 05 2013

There's an error in the last column of Riordan's table (change 46076 to 21076).
More terms from Vladeta Jovovic, Apr 16 2000
Link added and cross-references edited by Johannes W. Meijer, Aug 17 2009
Discussion of Riordan's definition of central factorial numbers added 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).

A156769 a(n) = denominator(2^(2n-2)/factorial(2n-1)).

Original entry on oeis.org

1, 3, 15, 315, 2835, 155925, 6081075, 638512875, 10854718875, 1856156927625, 194896477400625, 49308808782358125, 3698160658676859375, 1298054391195577640625, 263505041412702261046875, 122529844256906551386796875, 4043484860477916195764296875

Offset: 1

Johannes W. Meijer, Feb 15 2009

Resembles A036279, the denominators in the Taylor series for tan(x). The first difference occurs at a(12).
The numerators of the two formulas for this sequence lead to A001316, Gould's sequence.
Stephen Crowley indicated on Aug 25 2008 that a(n) = denominator(Zeta(2*n)/Zeta(1-2*n)) and here numerator((Zeta(2*n)/Zeta(1-2*n))/(2*(-1)^(n)*(Pi)^(2*n))) leads to Gould's sequence.
This sequence appears in the Eta and Zeta triangles A160464 and A160474. Its resemblance to the sequence of the denominators of the Taylor series for tan(x) led to the conjecture A156769(n) = A036279(n)*A089170(n-1). - Johannes W. Meijer, May 24 2009

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A036279 Denominators in Taylor series for tan(x).
Cf. A001316 Gould's sequence appears in the numerators.
Cf. A000265, A036279, A089170, A117972, A160464, A160469 (which resembles the numerators of the Taylor series for tan(x)), A160474. - Johannes W. Meijer, May 24 2009

[Denominator(4^(n-1)/Factorial(2*n-1)): n in [1..25]]; // G. C. Greubel, Jun 19 2021

a := n ->(2*n-1)!*2^(add(i,i=convert(n-1,base,2))-2*n+2); # Peter Luschny, May 02 2009

a[n_] := Denominator[4^(n-1)/(2n-1)!];
Array[a, 15] (* Jean-François Alcover, Jun 20 2018 *)

[denominator(4^(n-1)/factorial(2*n-1)) for n in (1..25)] # G. C. Greubel, Jun 19 2021

a(n) = denominator( Product_{k=1..n-1} 2/(k*(2*k+1)) ).
G.f.: (1/2)*z^(1/2)*sinh(2*z^(1/2)).
From Johannes W. Meijer, May 24 2009: (Start)
a(n) = abs(A117972(n))/A000265(n).
a(n) = A036279(n)*A089170(n-1). (End)
a(n) = A049606(2*n-1). - Zhujun Zhang, May 29 2019

A160480 The Beta triangle read by rows.

Original entry on oeis.org

-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

Offset: 2

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

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.

			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.

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.

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).

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

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 *)

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)!.

A002196 Denominators of coefficients for numerical integration.

Original entry on oeis.org

1, 12, 720, 60480, 3628800, 95800320, 2615348736000, 4483454976000, 32011868528640000, 51090942171709440000, 152579284313702400000, 120866571766215475200000, 50814724101952310083584000000

Offset: 0

N. J. A. Sloane

The denominators of these coefficients for numerical integration are a combination of the Bernoulli numbers B{2k}, the central factorial numbers A008955(n, k) and the factor (2n+1)!. - Johannes W. Meijer, Jan 27 2009

These numbers are the denominators of the constant term in the Laurent expansion of the even powers of the hyperbolic cosecant cosech^(2n)(x)/2^(2n) function. - Istvan Mezo, Apr 21 2023

			a(1) = 12 because (1/3)*int(t*(t^2-1^2), t=0..1) = -1/12.
a(3) = denom((-((1/6)/2)*(4) +((-1/30)/4)*(5) - ((1/42)/6)*(1))/5!) so a(3) = 60480. - _Johannes W. Meijer_, Jan 27 2009

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.
H. E. Salzer, Coefficients for numerical integration with central differences, Phil. Mag., 35 (1944), 262-264. [Annotated scanned copy]
H. E. Salzer, XXXII. Coefficients for numerical integration with central differences, Phil. Mag., 35 (1944), 262-264.
H. E. Salzer, Coefficients for repeated integration with central differences, Journal of Mathematics and Physics, 28 (1949), 54-61.
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 545.

Cf. A002195.
See A000367, A006954, A008955 and A009445 for underlying sequences.
Factor of ZS1[ -1,n] matrix coefficients in A160474.

a := n->denom((2/(2*n+1)!)*int(t*product(t^2-k^2,k=1..n),t=0..1)): seq(a(n), n=0..14); # Emeric Deutsch, Feb 20 2005
nmax:=12: with(combinat): A008955 := 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)) * A008955(n-1,n-k1), k1=1..n)/(2*n-1)! end do: a := n-> denom(Omega(n)): seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 27 2009, Revised Sep 21 2012

a[0] = 1; a[n_] := Sum[Binomial[2*n+k-1, 2*n-1]*Sum[Binomial[k, j]*Sum[(2*i-j)^(2*n+j)*Binomial[j, i]*(-1)^(-i), {i, 0, j/2}]/(2^j*(2*n+j)!), {j, 1, k}], {k, 1, 2*n}]/2^(2*n-1); Table[a[n] // Denominator, {n, 0, 12}] (* Jean-François Alcover, Apr 18 2014, after Vladimir Kruchinin *)
a[n_] := Denominator[SeriesCoefficient[1/2^(2*n)*Csch[x]^(2*n), {x, 0, 0}]] (* Istvan Mezo, Apr 21 2023 *)

a(n) = denominator of (2/(2*n+1)!)*int(t*product(t^2-k^2, k=1..n), t=0..1). - Emeric Deutsch, Jan 25 2005

a(0) = 1; a(n) = denominator [sum((-1)^(k+n+1) * (B{2k}/(2*k)) * A008955(n-1, n-k), k = 1..n) / (2*n-1)!] for n >= 1. - Johannes W. Meijer, Jan 27 2009

More terms from Emeric Deutsch, Jan 25 2005

Edited by Johannes W. Meijer, Sep 21 2012

cp's OEIS Frontend

A160479 The ZL(n) sequence of the Zeta and Lambda triangles A160474 and A160487.

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A160478 The p(n) sequence that is associated with the Zeta triangle A160474.

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A160475 First left hand column of the Zeta triangle A160474.

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A160477 Row sums of the Zeta triangle A160474.

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A008955 Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.

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A160464 The Eta triangle.

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A160487 The Lambda triangle.

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A156769 a(n) = denominator(2^(2*n-2)/factorial(2*n-1)).

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A156769 a(n) = denominator(2^(2n-2)/factorial(2n-1)).