A162443 -id:A162443 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

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

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

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812. [From _Johannes W. Meijer_, Jun 18 2009]
R. H. Boels, T. Hansen, String theory in target space, arXiv preprint arXiv:1402.6356 [hep-th], 2014.
T. L. Curtright, D. B. Fairlie, C. K. Zachos, A compact formula for rotations as spin matrix polynomials, arXiv preprint arXiv:1402.3541 [math-ph], 2014.
T. L. Curtright, T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arXiv:1408.0767 [math-ph], 2014.
M. Eastwood and H. Goldschmidt, Zero-energy fields on complex projective space, arXiv preprint arXiv:1108.1602 [math.DG], 2011.
M. Eastwood, The X-ray transform on projective space. - From _N. J. A. Sloane_, Oct 22 2012

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

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

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

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

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

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

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

More terms from Vladeta Jovovic, Apr 16 2000

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

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)!.

A055786 Numerators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).

Original entry on oeis.org

1, 1, 3, 5, 35, 63, 231, 143, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 9694845, 100180065, 116680311, 2268783825, 1472719325, 34461632205, 67282234305, 17534158031, 514589420475, 8061900920775, 5267108601573

Offset: 0

N. J. A. Sloane, Jul 13 2000

Note that the sequence is not monotonic.

			arcsin(x) is usually written as x + x^3/(2*3) + 1*3*x^5/(2*4*5) + 1*3*5*x^7/(2*4*6*7) + ..., which is x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 + ... (A055786/A002595) when reduced to lowest terms.
arccos(x) = Pi/2 - (x + (1/6)*x^3 + (3/40)*x^5 + (5/112)*x^7 + (35/1152)*x^9 + (63/2816)*x^11 + ...) (A055786/A002595).
arccsc(x) = 1/x + 1/(6*x^3) + 3/(40*x^5) + 5/(112*x^7) + 35/(1152*x^9) + 63/(2816*x^11) + ... (A055786/A002595).
arcsec(x) = Pi/2 -(1/x + 1/(6*x^3) + 3/(40*x^5) + 5/(112*x^7) + 35/(1152*x^9) + 63/(2816*x^11) + ...) (A055786/A002595).
arcsinh(x) = x - (1/6)*x^3 + (3/40)*x^5 - (5/112)*x^7 + (35/1152)*x^9 - (63/2816)*x^11 + ... (A055786/A002595).
i*Pi/2 - arccosh(x) = i*x + (1/6)*i*x^3 + (3/40)*i*x^5 + (5/112)*i*x^7 + (35/1152)*i*x^9 + (63/2816)*i*x^11 + (231/13312)*i*x^13 + (143/10240)*i*x^15 + (6435/557056)*i*x^17 + ... (A055786/A002595).
0, 1, 0, 1/6, 0, 3/40, 0, 5/112, 0, 35/1152, 0, 63/2816, 0, 231/13312, 0, 143/10240, 0, 6435/557056, 0, 12155/1245184, 0, 46189/5505024, 0, ... = A055786/A002595.
a(4) = 35 = 3*5*7*9 / gcd( 3*5*7*9, (2*4*6*8) * (2*4+1))

Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th German ed. 1965, ch. 4.2.6
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, Chap. 3.

T. D. Noe, Table of n, a(n) for n=0..200
Eric Weisstein's World of Mathematics, Inverse Cosecant.
Eric Weisstein's World of Mathematics, Inverse Cosine.
Eric Weisstein's World of Mathematics, Inverse Secant.
Eric Weisstein's World of Mathematics, Inverse Sine.
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosecant.
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosine.
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Sine.
Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1994. See p. 54.

Cf. A002595.
a(n) / A002595(n) = A001147(n) / ( A000165(n) * (2*n+1))
Cf. A162443 where BG1[-3,n] = (-1)*A002595(n-1)/A055786(n-1) for n >= 1. - Johannes W. Meijer, Jul 06 2009

[Numerator( (n+1)*Binomial(2*n+2,n+1)/(2^(2*n+1)*(2*n+1)^2) ): n in [0..25]]; // G. C. Greubel, Jan 25 2020

seq( numer( (n+1)*binomial(2*n+2,n+1)/(2^(2*n+1)*(2*n+1)^2) ), n=0..25); # G. C. Greubel, Jan 25 2020

Numerator/@Select[CoefficientList[Series[ArcSin[x],{x,0,60}],x], #!=0&]  (* Harvey P. Dale, Apr 29 2011 *)

vector(25, n, numerator(2*n*binomial(2*n,n)/(4^n*(2*n-1)^2)) ) \\ G. C. Greubel, Jan 25 2020

[numerator( (n+1)*binomial(2*n+2,n+1)/(2^(2*n+1)*(2*n+1)^2) ) for n in (0..25)] # G. C. Greubel, Jan 25 2020

a(n) / A052469(n) = A001147(n) / ( A000165(n) *2*n ). E.g., a(6) = 77 = 1*3*5*7*9*11 / gcd( 1*3*5*7*9*11, 2*4*6*8*10*12*12 ).

a(n) = numerator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))). - Johannes W. Meijer, Jul 06 2009

Edited by Johannes W. Meijer, Jul 06 2009

A162440 The pg(n) sequence that is associated with the Eta triangle A160464.

Original entry on oeis.org

2, 16, 144, 4608, 115200, 4147200, 203212800, 26011238400, 2106910310400, 210691031040000, 25493614755840000, 3671080524840960000, 620412608698122240000, 121600871304831959040000

Offset: 2

Johannes W. Meijer, Jul 06 2009

The EG1 matrix coefficients are defined by EG1[2m-1,1] = 2*eta(2m-1) and the recurrence relation EG1[2m-1,n] = EG1[2m-1,n-1] - EG1[2m-3,n-1]/(n-1)^2 with m = .., -2, -1, 0, 1, 2, ... and n = 1, 2, 3, ... . As usual, eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function. For the EG2 matrix, the even counterpart of the EG1 matrix, see A008955.
The coefficients in the columns of the EG1 matrix, for m >= 1 and n >= 2, can be generated with GFE(z;n) = ((-1)^(n-1)*r(n)*CFN1(z,n)*GFE(z;n=1) + ETA(z,n))/pg(n) for n >= 2.
The CFN1(z,n) polynomials depend on the central factorial numbers A008955 and the ETA(z,n) are the Eta polynomials which led to the Eta triangle, see for both A160464.
The pg(n) sequence can be generated with the first Maple program and the EG1[2m-1,n] matrix coefficients can be generated with the second Maple program.
The EG1 matrix is related to the ES1 matrix, see A160464 and the formulas below.

			The first few generating functions GFE(z;n) are:
GFE(z;n=2) = ((-1)*2*(z^2 - 1)*GFE(z;n=1) + (-1))/2,
GFE(z;n=3) = ((+1)*4*(z^4 - 5*z^2 + 4)*GFE(z;n=1) + (-11 + 2*z^2))/16,
GFE(z;n=4) = ((-1)*4*(z^6-14*z^4+49*z^2-36)*GFE(z;n=1) + (-114+29*z^2-2*z^4))/144.

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.

The ETA(z, n) polynomials and the ES1 matrix lead to the Eta triangle A160464.
The CFN1(z, n), the t1(n, m) and the EG2 matrix lead to A008955.
The EG1[ -1, n] equal (1/2)*A001803(n-1)/A046161(n-1).
The r(n) sequence equals A062383(n) (n>=1).
The e(n) sequence equals A029837(n) (n>=1).
Cf. A160473 (p(n) sequence).
Cf. A162443 (BG1 matrix), A162446 (ZG1 matrix) and A162448 (LG1 matrix).

nmax := 16; seq((n-1)!^2*2^floor(ln(n-1)/ln(2)+1), n=2..nmax);
# End program 1
nmax1 := 5; coln := 4; mmax1 := nmax1: for n from 0 to nmax1 do t1(n, 0) := 1 end do: for n from 0 to nmax1 do t1(n, n) := (n!)^2 end do: for n from 1 to nmax1 do for m from 1 to n-1 do t1(n, m) := t1(n-1, m-1)*n^2 + t1(n-1, m) end do: end do: for m from 1 to mmax1 do EG1[1-2*m, 1] := evalf((2^(2*m)-1)* bernoulli(2*m)/(m)) od: EG1[1, 1] := evalf(2*ln(2)): for m from 2 to mmax1 do EG1[2*m-1, 1] := evalf(2*(1-2^(1-(2*m-1))) * Zeta(2*m-1)) od: for m from -mmax1+coln to mmax1 do EG1[2*m-1, coln]:= (-1)^(coln+1)*sum((-1)^k*t1(coln-1, k) * EG1[1-2*coln+2*m+2*k, 1], k=0..coln-1)/(coln-1)!^2 od;
# End program 2 (Edited by Johannes W. Meijer, Sep 21 2012)

pg(n) = (n-1)!^2*2^floor(log(n-1)/log(2)+1) for n >= 2.
r(n) = 2^e(n) = 2^floor(log(n-1)/log(2)+1) for n >= 2.
EG1[ -1,n] = 2^(1-2*n)*(2*n-1)!/((n-1)!^2) for n >= 1.
GFE(z;n) = sum (EG1[2*m-1,n]*z^(2*m-2), m=1..infinity).
GFE(z;n) = (1-z^2/(n-1)^2)*GFE(z;n-1)-EG1[ -1,n-1]/(n-1)^2 for n = >2. with GFE(z;n=1) = 2*log(2)-Psi(z)-Psi(-z)+Psi(z/2)+Psi(-z/2) and Psi(z) is the digamma function.
EG1[2m-1,n] = (2*2^(1-2*n)*(2*n-1)!/((n-1)!^2)) * ES1[2m-1,n].

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]

A002595 Denominators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).

Original entry on oeis.org

1, 6, 40, 112, 1152, 2816, 13312, 10240, 557056, 1245184, 5505024, 12058624, 104857600, 226492416, 973078528, 2080374784, 23622320128, 30064771072, 635655159808, 446676598784, 11269994184704, 23639499997184, 6597069766656

Offset: 0

N. J. A. Sloane

arcsin(x) is usually written as x + x^3/(2*3) + 1*3*x^5/(2*4*5) + 1*3*5*x^7/(2*4*6*7) + ..., = x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 + ... when reduced to lowest terms.
arccos(x) = Pi/2 - (x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 + ...).
arccsc(x) = 1/x+1/(6*x^3)+3/(40*x^5)+5/(112*x^7)+35/(1152*x^9)+63/(2816*x^11)+...
arcsec(x) = Pi/2 -(1/x+1/(6*x^3)+3/(40*x^5)+5/(112*x^7)+35/(1152*x^9)+63/(2816*x^11)+...)
arcsinh(x) = x-1/6*x^3+3/40*x^5-5/112*x^7+35/1152*x^9-63/2816*x^11+...
arccsc(x) = arcsin(1/x) and arcsec(x) = arccos(1/x): 1 < |x|
arccsch(x) = arcsinh(1/x) for 1 < |x|
Also denominator of (2n-1)!! / ((2n+1)*(2n)!!) (n=>0).

W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
H. B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan, NY, 1968, Chap. 3.
Focus, vol. 16, no. 5, page 32, Oct 1996.
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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 31, equation 31:6:1 at page 290.

T. D. Noe, Table of n, a(n) for n=0..200
H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.
Eric Weisstein's World of Mathematics, Inverse Cosecant.
Eric Weisstein's World of Mathematics, Inverse Cosine.
Eric Weisstein's World of Mathematics, Inverse Secant.
Eric Weisstein's World of Mathematics, Inverse Sine.
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosecant.
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Sine.
Eric Weisstein's World of Mathematics, Archimedes' Spiral.
Herbert S. Wilf, Generatingfunctionology, Academic Press, NY, 1994. See p. 54.

A055786(n) / a(n) = A001147(n) / ( A000165(n) * (2*n+1))
Cf. A162443 where BG1[ -3,n] = (-1)*A002595(n-1)/A055786(n-1) for n =>1. - Johannes W. Meijer, Jul 06 2009
a(n) = 2*A143582(n+1) for n>=1. - Filip Zaludek, Oct 25 2016

Denominator[Take[CoefficientList[Series[ArcSin[x],{x,0,50}],x],{2,-1,2}]] (* Harvey P. Dale, Aug 06 2012 *)

a(n) = denominator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))); \\ Stefano Spezia, Dec 31 2024

a(n) = denominator((2*n)!/(2^(2*n)*(n)!^2*(2*n+1))). - Johannes W. Meijer, Jul 06 2009

A162444 Denominators of the BG1[ -5,n] coefficients of the BG1 matrix.

Original entry on oeis.org

1, 1, 3, 5, 35, 9, 231, 143, 6435, 12155, 3553, 88179, 96577, 1300075, 5014575, 102051, 100180065, 116680311, 2268783825, 210388475, 6892326441, 67282234305, 17534158031, 39583801575, 8061900920775, 169906729083

Offset: 1

Johannes W. Meijer, Jul 06 2009

For the numerators of the BG1[ -5,n] coefficients see A162443.

We observe that BG1[ -3,n] = (-1)*A002595(n-1)/A055786(n-1), i.e. they equal the inverted coefficients of the series expansion of arcsin(x), and that BG1[ -1,n] = A046161(n-1)/A001790(n-1), i.e. they equal the inverted coefficients of the series expansion of 1/sqrt(1-x).

			The first few formulas for the BG1[1-2*m,n] matrix coefficients are:
BG1[ -1,n] = (1)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -3,n] = (1-2*n)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -7,n] = (1-2*n+60*n^2-120*n^3)*4^(n-1)*(n-1)!^2/(2*n-2)!

A162443 are the numerators of the BG1[ -5, n] matrix coefficients.
The BG1[ -3, n] equal A002595(n-1)/A055786(n-1) for n =>1.
The BG1[ -1, n] equal A046161(n-1)/A001790(n-1) for n =>1.

a(n) = denom(BG1[ -5,n]) and A162443(n) = numer(BG1[ -5,n]) with BG1[ -5,n] = 4^(n-1)*(1-8*n+12*n^2)*(n-1)!^2/ (2*n-2)!.

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

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A160480 The Beta triangle read by rows.

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A055786 Numerators of Taylor series expansion of arcsin(x). Also arises from arccos(x), arccsc(x), arcsec(x), arcsinh(x).

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A162440 The pg(n) sequence that is associated with the Eta triangle A160464.

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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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