A002196 -id:A002196 - cp's OEIS frontend

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

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

A141417 (-1)^(n+1)A091137(n)a(0,n), where a(i,j) = Integral_{x=i..i+1} x(x-1)(x-2)...(x-j+1)/j! dx.

Original entry on oeis.org

-1, 1, 1, 1, 19, 27, 863, 1375, 33953, 57281, 3250433, 5675265, 13695779093, 24466579093, 132282840127, 240208245823, 111956703448001, 205804074290625, 151711881512390095, 281550972898020815, 86560056264289860203, 161867055619224199787, 20953816286242674495191, 39427936010479474495191

Offset: 0

Paul Curtz, Aug 05 2008

sign

This is row i=0 of an array defined as T(i,j) = (-1)^(i+j+1)*A091137(j)*a(i,j), columns j >= 0, which starts
  -1,   1,   1,    1,    19,    27,   863, ...
   1,  -3,   5,    1,    11,    11,   271, ...
  -1,   5, -23,    9,    19,    11,   191, ...
   1,  -7,  53,  -55,   251,    27,   271, ...
  -1,   9, -95,  161, -1901,   475,   863, ...
   1, -11, 149, -351,  6731, -4277, 19087, ...
  ...
The first two rows are related via T(0,j) = A027760(j)*T(0,j-1) - T(1,j).

P. Curtz, Integration .., note 12, C.C.S.A., Arcueil, 1969.

Cf. A141047, A140811, A140825.

A091137 := proc(n) local a, i, p ; a := 1 ; for i from 1 do p := ithprime(i) ; if p > n+1 then break; fi; a := a*p^floor(n/(p-1)) ; od: a ; end proc:
A048994 := proc(n, k) combinat[stirling1](n, k) ; end proc:
a := proc(i,j) add(A048994(j,k)*x^k,k=0..j) ; int(%,x=i..i+1) ; %/j! ; end proc:
A141417 := proc(n) (-1)^(n+1)*A091137(n)*a(0,n) ; end proc:
seq(A141417(n),n=0..40) ; # R. J. Mathar, Nov 17 2010

(* a7 = A091137 *) a7[n_] := a7[n] = Times @@ Select[ Divisors[n]+1, PrimeQ]*a7[n-1]; a7[0]=1; a[n_] := (-1)^(n+1) * a7[n] * Integrate[ (-1)^n*Pochhammer[-x, n], {x, 0, 1}]/n!; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Aug 10 2012 *)

a(n):=if n=0 then -1 else num(n*(n+1)*sum(((-1)^(n-k)*stirling2(n+k,k)*binomial(2*n-1,n-k))/((n+k)*(n+k-1)),k,1,n)); /* Vladimir Kruchinin, Dec 12 2016 */

a(i,j) = a(i-1,j) + a(i-1,j-1), see reference page 33.
(q+1-j)*Sum_{j=0..q} a(i,j)*(-1)^(q-j) = binomial(i,q), see reference page 35.
a(n) = numerator(n*(n+1)*Sum_{k=1..n} ((-1)^(n-k)*Stirling2(n+k,k)*binomial(2*n-1,n-k))/((n+k)*(n+k-1))), n>0, a(0)=-1. - Vladimir Kruchinin, Dec 12 2016

Erroneous formula linking A091137 and A002196 removed, and more terms and program added by R. J. Mathar, Nov 17 2010

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).

A002195 Numerators of coefficients for numerical integration.

Original entry on oeis.org

1, -1, 11, -191, 2497, -14797, 92427157, -36740617, 61430943169, -23133945892303, 16399688681447, -3098811853954483, 312017413700271173731, -69213549869569446541, 53903636903066465730877, -522273861988577772410712439, 644962185719868974672135609261

Offset: 0

N. J. A. Sloane

The numerators 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 numerators of the constant term in the Laurent expansion of the cosech^(2n)(x)/2^(2n) function. - Istvan Mezo, Apr 21 2023

			a(1) = -1 because (1/3)*int(t*(t^2-1^2),t=0..1) = -1/12.
a(3) = numer((-((1/6)/2)*(4) +((-1/30)/4)*(5) - ((1/42)/6)*(1))/5!) so a(3) = -191. - _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. A002196.

See A000367, A006954, A008955 and A009445 for underlying sequences.

a:=n->numer((2/(2*n+1)!)*int(t*product(t^2-k^2,k=1..n),t=0..1)): seq(a(n),n=0..16); # Emeric Deutsch, Feb 20 2005
nmax:=16: 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-> numer(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}]; Table[a[n] // Numerator, {n, 0, 16}] (* Jean-François Alcover, Apr 18 2014, after Vladimir Kruchinin *)
a[n_] := Numerator[SeriesCoefficient[1/2^(2*n)*Csch[x]^(2*n), {x, 0, 0}]] (* Istvan Mezo, Apr 21 2023 *)

a(n):=num(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)); /* Vladimir Kruchinin, Feb 04 2013 */

a(n) = numerator of (2/(2*n+1)!)*Integral_{t=0..1} t*Product_{k=1..n} t^2-k^2. - Emeric Deutsch, Jan 25 2005
a(0) = 1; a(n) = numerator [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
a(n) = numerator(sum(k=1..2*n, binomial(2*n+k-1,2*n-1)*sum(j=1..k, (binomial(k,j)*sum(i=0,j/2, (2*i-j)^(2*n+j)*binomial(j,i)*(-1)^(-i)))/(2^j*(2*n+j)!)))), n>0, a(0)=1. - Vladimir Kruchinin, Feb 04 2013

More terms from Emeric Deutsch, Jan 25 2005

Edited by Johannes W. Meijer, Sep 21 2012

A162446 Numerators of the column sums of the ZG1 matrix.

Original entry on oeis.org

-13, 401, -68323, 2067169, -91473331, 250738892357, -12072244190753, 105796895635531, -29605311573467996893, 9784971385947359480303, -5408317625058335310276319, 2111561851139130085557412009

Offset: 2

Johannes W. Meijer, Jul 06 2009

The ZG1 matrix coefficients are defined by ZG1[2m-1,1] = 2*zeta(2m-1) for m = 2, 3, .. , and the recurrence relation ZG1[2m-1,n] = (ZG1[2m-3,n-1] - (n-1)^2*ZG1[2m-1,n-1])/(n*(n-1)) with m = .. , -2, -1, 0, 1, 2, .. and n = 1, 2, 3, .. , under the condition that n <= (m-1). As usual zeta(m) is the Riemann zeta function. For the ZG2 matrix, the even counterpart of the ZG1 matrix, see A008955.
These two formulas enable us to determine the values of the ZG1[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, ZG1[1,1] = 2*gamma, with gamma the Euler-Mascheroni constant, we can determine them all.
The coefficients in the columns of the ZG1 matrix, for m >= 1 and n >= 2, can be generated with GFZ(z;n) = (hg(n)*CFN1(z;n)*GFZ(z;n=1) + ZETA(z;n))/pg(n) with pg(n) = 6*(n-1)!* (n)!*A160476(n) and hg(n) = 6*A160476(n). For the CFN1(z;n) and the ZETA(z;n) polynomials see A160474.
The column sums cs(n) = sum(ZG1[2*m-1,n], m = 1 .. infinity), for n >= 2, of the ZG1 matrix can be determined with the first Maple program. In this program we have made use of the remarkable fact that if we take ZGx[2*m-1,n] = 2, for m >= 1, and ZGx[ -1,n] = ZG1[ -1,n] and assume that the recurrence relation remains the same we find that the column sums of this new matrix converge to the same values as the original cs(n).
The ZG1[2*m-1,n] matrix coefficients can be generated with the second Maple program.
The ZG1 matrix is related to the ZS1 matrix, see A160474 and the formulas below.

			The first few generating functions GFZ(z;n) are:
GFZ(z;2) = (6*(1*z^2-1)*GFZ(z;1) + (-1))/12
GFZ(z;3) = (60*(z^4-5*z^2+4)*GFZ(z;1) + (51-10*z^2))/720
GFZ(z;4) = (1260*(z^6-14*z^4+49*z^2-36)*GFZ(z;1) + (-10594+2961*z^2-210*z^4))/181440

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.

See A162447 for the denominators of the column sums.
The pg(n) and hg(n) sequences lead to A160476.
The ZG1[ -1, n] coefficients lead to A000984, A002195 and A002196.
The ZETA(z, n) polynomials and the ZS1 matrix lead to the Zeta triangle A160474.
The CFN1(z, n), the cfn1(n, k) and the ZG2 matrix lead to A008955.
The b(n) sequence equals A001790(n)/ A120777(n-1) for n >= 1.
Cf. A001620 (gamma) and A010790 (n!*(n+1)!).
Cf. A162440 (EG1 matrix), A162443 (BG1 matrix) and A162448 (LG1 matrix)

nmax := 13; mmax := nmax: with(combinat): cfn1 := proc(n, k): sum((-1)^j1*stirling1(n+1, n+1-k+j1)*stirling1(n+1, n+1-k-j1), j1=-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)! od: for n from 1 to nmax do ZG1[ -1, n] := binomial(2*n, n)*Omega(n) od: for n from 1 to nmax do ZGx[ -1, n] := ZG1[ -1, n] od: for m from 1 to mmax do ZGx[2*m-1, 1] := 2 od: for n from 2 to nmax do for m from 1 to mmax do ZGx[2*m-1, n] := (((ZGx[2*m-3, n-1]-(n-1)^2*ZGx[2*m-1, n-1])/(n*(n-1)))) od; s(n) := 0: for m from 1 to mmax do s(n) := s(n) + ZGx[2*m-1, n] od: od: seq(s(n), n=2..nmax);
# End program 1
nmax1 := 5; ncol := 3; Digits := 20: mmax1 := nmax1: with(combinat): cfn1 := proc(n, k): sum((-1)^j1*stirling1(n+1, n+1-k+j1)*stirling1(n+1, n+1-k-j1), j1=-k..k) end proc: ZG1[1, 1] := evalf(2*gamma): for m from 1 to mmax1 do ZG1[1-2*m, 1] := -bernoulli(2*m)/m od: for m from 2 to mmax1 do ZG1[2*m-1, 1] := evalf(2*Zeta(2*m-1)) od: for n from 1 to nmax1 do for m from -mmax1 to mmax1 do ZG1[2*m-1, n] := sum((-1)^(k1+1)*cfn1(n-1, k1-1)*ZG1[2*m-(2*n-2*k1+1), 1] /((n-1)!*(n)!), k1=1..n) od; od; for m from -mmax1+ncol to mmax1 do ZG1[2*m-1, ncol] := ZG1[2*m-1, ncol] od;
# End program 2
# Maple programs edited by Johannes W. Meijer, Sep 25 2012

a(n) = numer(cs(n)) and denom(cs(n)) = A162447(n).
with cs(n) = sum(ZG1[2*m-1,n], m = 1 .. infinity) for n >= 2.
GFZ(z;n) = sum( ZG1[2*m-1,n]*z^(2*m-2),m=1..infinity)
GFZ(z;n) = ZG1[ -1,n-1]/(n*(n-1))+(z^2-(n-1)^2)*GFZ(z;n-1)/(n*(n-1)) for n >= 2 with GFZ(z;n=1) = -Psi(1+z) - Psi(1-z).
ZG1[ -1,n] = binomial(2*n,n)*Omega[n] = A000984(n)*A002195(n)/A002196(n).
ZG1[2*m-1,n] = b(n)*ZS1[2*m-1,n] with b(n) = binomial(2*n,n)/2^(2*n-1) for n >= 1.

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

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A141417 (-1)^(n+1)*A091137(n)*a(0,n), where a(i,j) = Integral_{x=i..i+1} x*(x-1)*(x-2)*...*(x-j+1)/j! dx.

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A160474 The Zeta triangle.

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A002195 Numerators of coefficients for numerical integration.

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A162446 Numerators of the column sums of the ZG1 matrix.

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A002682 Denominators of coefficients for repeated integration.

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A141417 (-1)^(n+1)A091137(n)a(0,n), where a(i,j) = Integral_{x=i..i+1} x(x-1)(x-2)...(x-j+1)/j! dx.