A162440 -id:A162440 - cp's OEIS frontend

A062383 a(0) = 1: for n>0, a(n) = 2^floor(log_2(n)+1) or a(n) = 2*a(floor(n/2)).

1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128

Offset: 0

Antti Karttunen, Jun 19 2001

Informally, write down 1 followed by 2^k 2^(k-1) times, for k = 1,2,3,4,... These are the denominators of the binary van der Corput sequence (see A030101 for the numerators). - N. J. A. Sloane, Dec 01 2019
a(n) is the denominator of the form 2^k needed to make the ratio (2n-1)/2^k lie in the interval [1-2], i.e. such ratios are 1/1, 3/2, 5/4, 7/4, 9/8, 11/8, 13/8, 15/8, 17/16, 19/16, 21/16, ... where the numerators are A005408 (The odd numbers).
Let A_n be the upper triangular matrix in the group GL(n,2) that has zero entries below the diagonal and 1 elsewhere. For example for n=4 the matrix is / 1,1,1,1 / 0,1,1,1 / 0,0,1,1 / 0,0,0,1 /. The order of this matrix as an element of GL(n,2) is a(n-1). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 14 2001
A006257(n)/a(n) = (0, 0.1, 0.01, 0.11, 0.001, ...) enumerates all binary fractions in the unit interval [0, 1). - Fredrik Johansson, Aug 14 2006
a(n) = maximum of row n+1 in A240769. - Reinhard Zumkeller, Apr 13 2014
This is the discriminator sequence for the odious numbers. - N. J. A. Sloane, May 10 2016
From Jianing Song, Jul 05 2025: (Start)
a(n) is the period of {binomial(N,n) mod 2: N in Z}. For the general result, see A349593.
Since the modulus (2) is a prime, the remainder of binomial(N,n) is given by Lucas's theorem. (End)

Harry J. Smith, Table of n, a(n) for n = 0..1000
L. K. Arnold, S. J. Benkoski and B. J. McCabe, The discriminator (a simple application of Bertrand's postulate). Amer. Math. Monthly 92 (1985), 275-277.
Sajed Haque, Chapter 2.6.1 of Discriminators of Integer Sequences, 2017, See p. 33.
S. Haque and J. Shallit, Discriminators and k-regular sequences, arXiv:1605.00092 [cs.DM], 2016.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Wikipedia, Lucas's theorem
Index to divisibility sequences

Apart from the initial term, equals 2 * A053644. MASKTRANSi(A062383) seems to give a signed form of A038712. (See identities at A053644). floor_log_2 given in A054429.
Equals A003817(n)+1. Cf. A002884.
Bisection of A065285. Cf. A076877.
Equals for n>=1 the r(n) sequence of A160464. - Johannes W. Meijer, May 24 2009
Equals the r(n) sequence of A162440 for n>=1. - Johannes W. Meijer, Jul 06 2009
Cf. A030101, A264618, A264619.
Discriminator of the odious numbers (A000069). - Jeffrey Shallit, May 08 2016
Column 2 of A349593. A064235 (if offset 0), A385552, A385553, and A385554 are respectively columns 3, 5, 6, and 10.

import Data.List (transpose)
a062383 n = a062383_list !! n
a062383_list = 1 : zs where
   zs = 2 : (map (* 2) $ concat $ transpose [zs, zs])
-- Reinhard Zumkeller, Aug 27 2014, Mar 13 2014

[2^Floor(Log(2,2*n+1)): n in [0..70]]; // Bruno Berselli, Mar 04 2016

[seq(2^(floor_log_2(j)+1),j=0..127)]; or [seq(coerce1st_octave((2*j)+1),j=0..127)]; or [seq(a(j),j=0..127)];
coerce1st_octave := proc(r) option remember; if(r < 1) then coerce1st_octave(2*r); else if(r >= 2) then coerce1st_octave(r/2); else (r); fi; fi; end;
A062383 := proc(n)
    option remember;
    if n = 0 then
        1 ;
    else
        2*procname(floor(n/2));
    end if;
end proc:
A062383 := n -> 1 + Bits:-Iff(n, n):
seq(A062383(n), n=0..69); # Peter Luschny, Sep 23 2019

a[n_] := a[n] = 2 a[n/2 // Floor]; a[0] = 1; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 04 2016 *)
Table[2^Floor[Log2[n] + 1], {n, 0, 20}] (* Eric W. Weisstein, Nov 17 2017 *)
2^Floor[Log2[Range[0, 20]] + 1] (* Eric W. Weisstein, Nov 17 2017 *)
2^BitLength[Range[0, 100]] (* Paolo Xausa, Jan 29 2025 *)

{ a=1; for (n=0, 1000, write("b062383.txt", n, " ", a*=ceil((n + 1)/a)) ) } \\ Harry J. Smith, Aug 06 2009

a(n)=1<<(log(2*n+1)\log(2)) \\ Charles R Greathouse IV, Dec 08 2011

def A062383(n): return 1 << n.bit_length() # Chai Wah Wu, Jun 30 2022

a(1) = 1 and a(n+1) = a(n)*ceiling(n/a(n)). - Benoit Cloitre, Aug 17 2002
G.f.: 1/(1-x) * (1 + Sum_{k>=0} 2^k*x^2^k). - Ralf Stephan, Apr 18 2003
a(n) = A142151(2*n)/2 + 1. - Reinhard Zumkeller, Jul 15 2008
log(a(n))/log(2) = A029837(n+1). - Johannes W. Meijer, Jul 06 2009
a(n+1) = a(n) + A099894(n). - Reinhard Zumkeller, Aug 06 2009
a(n) = A264619(n) - A264618(n). - Reinhard Zumkeller, Dec 01 2015
a(n) is the smallest power of 2 > n. - Chai Wah Wu, Nov 04 2016
a(n) = 2^ceiling(log_2(n+1)). - M. F. Hasler, Sep 20 2017

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).

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.

A162448 Numerators of the column sums of the LG1 matrix.

Original entry on oeis.org

-11, 863, -215641, 41208059, -9038561117, 28141689013943, -2360298440602051, 3420015713873670001, -147239749512798268300237, 176556159649301309969405807, -178564975300377173768513546347

Offset: 2

Johannes W. Meijer, Jul 06 2009

The LG1 matrix coefficients are defined by LG1[2m,1] = 2*lambda(2m+1) for m = 1, 2, .. , and the recurrence relation LG1[2*m,n] = LG1[2*m-2,n-1]/((2*n-3)*(2*n-1)) - (2*n-3)*LG1[2*m,n-1]/(2*n-1) with m = .. , -2, -1, 0, 1, 2, .. and n = 1, 2, 3, .. , under the condition that n <= m. As usual lambda(m) = (1-2^(-m))*zeta(m) with zeta(m) the Riemann zeta function. For the LG2 matrix, the even counterpart of the LG1 matrix, see A008956.
These two formulas enable us to determine the values of the LG1[2*m,n] coefficients, with m all integers and n all positive integers, but not for all. If we choose, somewhat but not entirely arbitrarily, LG1[0,1] = gamma, with gamma the Euler-Mascheroni constant, we can determine them all.
The coefficients in the columns of the LG1 matrix, for m >= 1 and n >= 2, can be generated with GFL(z;n) = (hg(n)*CFN2(z;n)*GFL(z;n=1) + LAMBDA(z;n))/pg(n) with pg(n) = 6*(2*n-3)!!*(2*n-1)!!*A160476(n) and hg(n) = 6*A160476(n). For the CFN2(z;n) and the LAMBDA(z;n) see A160487.
The values of the column sums cs(n) = sum(LG1[2*m,n], m = 0.. infinity), for n >= 2, can be determined with the first Maple program. In this program we have made use of the remarkable fact that if we take LGx[2*m,n] = 2, for m >= 0, and LGx[ -2,n] = LG1[ -2,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 LG1[2*m,n] matrix coefficients can be generated with the second Maple program.
The LG1 matrix is related to the LS1 matrix, see A160487 and the formulas below.

			The first few generating functions GFL(z;n) are:
GFL(z;2) = (6*(z^2-1)*GFL(z;1)+(1))/18
GFL(z;3) = (60*(z^4-10*z^2+9)*GFL(z;1)+(-107+10*z^2))/2700
GFL(z;4) = (1260*(z^6-35*z^4+259*z^2-225)*GFL(z;1)+(59845-7497*z^2+210*z^4))/ 1984500

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 A162449 for the denominators of the column sums.
The LAMBDA(z, n) polynomials and the LS1 matrix lead to the Lambda triangle A160487.
The CFN2(z, n), the cfn2(n, k) and the LG2 matrix lead to A008956.
The pg(n) and hg(n) sequences lead to A160476.
The LG1[ -2, n] lead to A002197, A002198, A061549 and A001790.
Cf. A001620 (gamma) and A079484 ((2n-1)!!*(2n+1)!!).
Cf. A162440 (EG1 matrix), A162443 (BG1 matrix) and A162446 (ZG1 matrix)

nmax := 12; mmax := nmax: 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), k1=1..n)/ (2*4^(n-1)*(2*n-1)!) od: for n from 1 to nmax do LG1[ -2, n] := (-1)^(n+1)*4*Delta(n-1)* 4^(2*n-2)/binomial(2*n-2, n-1) od: for n from 1 to nmax do LGx[ -2, n] := LG1[ -2, n] od: for m from 0 to mmax do LGx[2*m, 1] := 2 od: for n from 2 to nmax do for m from 0 to mmax do LGx[2*m, n] := LGx[2*m-2, n-1]/((2*n-3)*(2*n-1)) - (2*n-3)*LGx[2*m, n-1]/(2*n-1) od: od: for n from 2 to nmax do s(n) := 0; for m from 0 to mmax-1 do s(n) := s(n) + LGx[2*m, n] od: od: seq(s(n), n=2..nmax);
# End program 1
nmax1:=5; ncol:=3; Digits:=20: mmax1:=nmax1: 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 m from 1 to mmax1 do LG1[ -2*m, 1] := (((2^(2*m-1)-1)*bernoulli(2*m)/m)) od: LG1[0, 1] := evalf(gamma): for m from 2 to mmax1 do LG1[2*m-2, 1] := evalf(2*(1-2^(-2*m+1))*Zeta(2*m-1)) od: for m from -mmax1+ncol-1 to mmax1-1 do LG1[2*m, ncol] := sum((-1)^(k1+1)*cfn2(ncol-1, k1-1)* LG1[2*m-(2*ncol-2*k1), 1], k1=1..ncol)/(doublefactorial(2*ncol-3)*doublefactorial(2*ncol-1)) od;
# End program 2
# Maple programs edited by Johannes W. Meijer, Sep 25 2012

a(n) = numer(cs(n)) and denom(cs(n)) = A162449(n).
with cs(n) = sum(LG1[2*m,n], m = 0 .. infinity) for n >= 2.
GFL(z;n) = sum( LG1[2*m,n]*z^(2*m-2),m=1..infinity)
GFL(z;n) = (LG1[ -2,n-1])/((2*n-3)*(2*n-1))+(z^2/((2*n-3)*(2*n-1))-(2*n-3)/(2*n-1))*GFL(z;n-1) with GFL(z;n=1) = -2*Psi(1-z)+Psi(1-(z/2))-(Pi/2)*tan(Pi*z/2)
LG1[ -2,n] = (-1)^(n+1)*4*(A061549(n-1)/A001790(n-1))*(A002197(n-1)/A002198(n-1))
LG1[2*m,n] = (4^(n-1)/((2*n-1)*binomial(2*n-2,n-1)))*LS1[2*m,n]

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A062383 a(0) = 1: for n>0, a(n) = 2^floor(log_2(n)+1) or a(n) = 2*a(floor(n/2)).

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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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A162446 Numerators of the column sums of the ZG1 matrix.

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A162448 Numerators of the column sums of the LG1 matrix.

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A160473 The p(n) sequence that is associated with the Eta triangle A160464.

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