A008956 -id:A008956 - cp's OEIS frontend

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

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

Offset: 1

Johannes W. Meijer, Jul 06 2009

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

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

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.
J. M. Amigo, Relations among Sums of Reciprocal Powers Part II, International Journal of Mathematics and Mathematical Sciences , Volume 2008 (2008), pp. 1-20.

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

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

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

A160476 The first right hand column of the Zeta and Lambda triangles.

Original entry on oeis.org

1, 10, 210, 420, 4620, 60060, 60060, 2042040, 116396280, 581981400, 13385572200, 13385572200, 13385572200, 388181593800, 12033629407800, 24067258815600, 24067258815600, 890488576177200, 890488576177200

Offset: 2

Johannes W. Meijer, May 24 2009

This intriguing sequence makes its appearance in the Zeta and Lambda triangles.

The first Maple algorithm is related to the Zeta triangle and the second to the Lambda triangle. Both generate the sequence of the first right hand column of these triangles.

The Zeta and Lambda triangles are A160474 and A160487.
Cf. A160478 and A160490, A008955 and A008956, A048896 and A000265.
Appears in A162446 (ZG1 matrix) and A162448 (LG1 matrix) [Johannes W. Meijer, Jul 06 2009]

nmax := 20; 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: for n from 2 to nmax do a1(n) := p(n)*2^(2*n-3)/(3*factorial(2*n-1)) od: seq(a1(n), n=2..nmax);
# End first program (program edited, Johannes W. Meijer, Sep 20 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)!); 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 nmax1 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 nmax1 do b(n) := denom(Lcgz(n+1)) end do: for n from 1 to nmax1 do b(n) := 2*n*denom(Delta(n-1))/2^(2*n) end do: p(2) := b(1): for n from 2 to nmax1 do p(n+1) := lcm(p(n)*(2*n)*(2*n-1), b(n)) end do: for n from 2 to nmax1 do a2(n) := p(n)/(6*factorial(2*n-2)) od: seq(a2(n), n=2..nmax1);
# End second program (program edited, Johannes W. Meijer, Sep 20 2012)

a(n) = A160490(n)/(6*(2*n-2)!) for n = 2, 3, .. .
a(n) = A160478(n)*M(n) with M(n) = 2^(2*n-3)/(3*(2*n-1)!) for n=2, 3, .. .
M(n) = A048896(n-2)/(9*M1(n-1)) with M1(n) = (2*n+1)*A000265(n)*M1(n-1) for n = 2, 3, .. , and M1(1) = 1.
a(n+1)/a(n) = A160479(n+1) [Johannes W. Meijer, Oct 07 2009]

A161736 Denominators of the column sums of the BG2 matrix.

Original entry on oeis.org

1, 9, 75, 1225, 19845, 160083, 1288287, 41409225, 1329696225, 10667118605, 85530896451, 1371086188563, 21972535073125, 176021737014375, 1409850293610375, 90324408810638025, 5786075364399106425, 46326420401234675625, 370882277949065911875, 5938020471163465810125

Offset: 2

Johannes W. Meijer, Jun 18 2009

The BG2 matrix coefficients, see also A008956, are defined by 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, .. .
A different way to define the matrix coefficients is BG2[2*m,n] = (1/m)*sum(LAMBDA(2*m-2*k,n-1)*BG2[2*k,n], k=0..m-1) with LAMBDA(2*m,n-1) = (1-2^(-2*m))*zeta(2*m)-sum((2*k-1)^(-2*m), k=1..n-1) and BG2[0,n] = Pi/2 for m = 0, 1, 2, .. , and n = 1, 2, 3 .. , with zeta(m) the Riemann zeta function.
The columns sums of the BG2 matrix are defined by sb(n) = sum(BG2[2*m,n], m=0..infinity) for n = 2, 3, .. . For large values of n the value of sb(n) approaches Pi/2.
It is remarkable that if we assume that BG2[2m,1] = 2 for m = 0, 1, .. the columns sums of the modified matrix converge to the original sb(n) values. The first Maple program makes use of this phenomenon and links the sb(n) with the central factorial numbers A008956.
The column sums sb(n) can be linked to other sequences, see the second Maple program.
We observe that the column sums sb(n) of the BG2(n) matrix are related to the column sums sl(n) of the LG2(n) matrix, see A008956, by sb(n) = (-1)^(n+1)*(2*n-1)*sl(n).
a(n+2), for n >= 0, seems to coincide with the numerators belonging to A278145. - Wolfdieter Lang, Nov 16 2016
Suppose that, given values f(x-2*n+1), f(x-2*n+3), ..., f(x-1), f(x+1), ..., f(x+2*n-3), f(x+2*n-1), we approximate f(x) using the first 2*n terms of its Taylor series. Then 1/sb(n+1) is the coefficient of f(x-1) and f(x+1). - Matthew House, Dec 03 2024

			sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..

G. C. Greubel, Table of n, a(n) for n = 2..830
R. Arratia, S. Garibaldi, and J. Kilian, Asymptotic distribution for the birthday problem with multiple coincidences, via an embedding of the collision process, arXiv:1310.7055 [math.PR], 2013.

Cf. Numerators A161737.
Cf. A001818, A001902, A008956, A013777, A134372 and A134375.
Cf. A001147, A133221, A161738 and A002474, A278145.

[Denominator((2^(4*n-5)*(Factorial(n-1))^4)/((n-1)*(Factorial(2*n-2))^2)): n in [2..20]]; // G. C. Greubel, Sep 26 2018

nmax := 18; for n from 0 to nmax do A001818(n) := (doublefactorial(2*n-1))^2 od: for n from 0 to nmax do A008956(n, 0):=1 od: for n from 0 to nmax do A008956(n, n) := A001818(n) od: for n from 1 to nmax do for m from 1 to n-1 do A008956(n, m) := (2*n-1)^2*A008956(n-1, m-1) + A008956(n-1, m) od: od: for n from 1 to nmax do for m from 0 to n do s(n, m):=0; s(n, m) := s(n, m)+ sum((-1)^k1*A008956(n, n-k1), k1=0..n-m): od: sb1(n+1) := sum(s(n, k1), k1=1..n) * 2/A001818(n); od: seq(sb1(n), n=2..nmax); # End program 1
nmax1 := nmax; for n from 0 to nmax1 do A001147(n):= doublefactorial(2*n-1) od: for n from 0 to nmax1/2 do A133221(2*n+1) := A001147(n); A133221(2*n) := A001147(n) od: for n from 0 to nmax1 do A002474(n) := 2^(2*n+1)*n!*(n+1)! od: for n from 1 to nmax1 do A161738(n) := ((product((2*n-3-2*k1), k1=0..floor(n/2-1)))) od: for n from 2 to nmax1 do sb2(n) := A002474(n-2) / (A161738(n)*A133221(n-1))^2 od: seq(sb2(n), n=2..nmax1); # End program 2
# Above Maple programs edited by Johannes W. Meijer, Sep 25 2012
r := n -> (1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2: a := n -> numer(simplify(r(n))):
seq(a(n), n = 1..21);  # Peter Luschny, Feb 12 2025

sb[2]=2; sb[n_] := sb[n] = sb[n-1]*4*(n-1)*(n-2)/(2n-3)^2; Table[sb[n] // Denominator, {n, 2, 20}] (* Jean-François Alcover, Aug 14 2017 *)

{a(n) = if( n<2, 0, n--; numerator( binomial( 2*n, n)^2 * n / 2^(n+1) ))}; /* Michael Somos, May 09 2011 */

a(n) = denom(sb(n)) with sb(n) = (2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2) and A161737(n) = numer(sb(n)).
a(n+1) = numerator of C(2*n,n)^2 * n / 2^(n+1). - Michael Somos, May 09 2011
a(n) = A001902(2*n-3). - Mats Granvik, Nov 25 2018
a(n) = numerator((1/Pi)*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2). - Peter Luschny, Feb 13 2025

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]

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

Original entry on oeis.org

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

A002197 Numerators of coefficients for numerical integration.

Original entry on oeis.org

1, 17, 367, 27859, 1295803, 5329242827, 25198857127, 11959712166949, 11153239773419941, 31326450596954510807, 3737565567167418110609, 2102602044094540855003573, 189861334343507894443216783

Offset: 0

N. J. A. Sloane

nonn

The numerators of these coefficients for numerical integration are a combination of the Bernoulli numbers B{2k}, the central factorial numbers A008956(n, k) and the factor 4^n*(2*n+1)!. - Johannes W. Meijer, Jan 27 2009

			a(2) = numer(((1-2^1)*(-1)*((1/6)/2)*(9) + (1-2^3)*(1)*((-1/30)/4)*(10) + (1-2^5)*(-1)*((1/42)/6)*(1))/(2*4^2*5!)) so a(2) = 367. - _Johannes W. Meijer_, Jan 27 2009

H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218.
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).

T. D. Noe, Table of n, a(n) for n=0..100
H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218. [Annotated scanned copy]
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 545.

Cf. A002198.
See A000367, A006954, A008956 and A002671 for underlying sequences.
Factor of the LS1[-2,n] matrix coefficients in A160487.

nmax:=13: for n from 0 to nmax do A008956(n, 0) := 1: A008956(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax do for k from 1 to n-1 do A008956(n, k) := (2*n-1)^2*A008956(n-1, k-1) + A008956(n-1, k) od: od: for n from 0 to nmax do Delta(n) := sum((1-2^(2*k1-1)) * (-1)^(k1) * (bernoulli(2*k1)/(2*k1)) * A008956(n, n+1-k1), k1=1..n+1) / (2*4^(n)*(2*n+1)!) end do: a:=n-> numer(Delta(n)): seq(a(n), n=0..nmax-1); # Johannes W. Meijer, Jan 27 2009, revised Sep 21 2012

CoefficientList[Series[1/x - 1/Sqrt[x]/ArcSin[Sqrt[x]], {x, 0, 12}], x] // Numerator (* Jean-François Alcover, Jul 05 2011, after Vladeta Jovovic *)

a(n):=(sum(binomial(2*n+k-1,2*n-2)*sum((binomial(k+1,j)*sum((2*i-j)^(2*n+j)*binomial(j,i)*(-1)^(n-i),i,0,j/2))/(2^(j-1)*(2*n+j)!),j,1,k+1),k,0,2*n-1))/(2*n-1);
makelist(num(a(n)),n,0,10); /* Vladimir Kruchinin, May 16 2013 */

Numerators of coefficients in expansion of 1/x-1/sqrt(x)/arcsin(sqrt(x)). - Vladeta Jovovic, Aug 11 2002
a(n) = numerator [sum((1-2^(2*k-1)) * (-1)^(k) * (B{2k}/(2*k)) * A008956(n, n+1-k), k=1..n+1) / (2*4^(n)*(2*n+1)!)] for n >= 0. - Johannes W. Meijer, Jan 27 2009
a(n) = numerator((sum(k=0..2*n-1, binomial(2*n+k-1,2*n-2)*sum(j=1..k+1, (binomial(k+1,j)*sum(i=0..j/2,(2*i-j)^(2*n+j)*binomial(j,i)*(-1)^(n-i)))/(2^(j-1)*(2*n+j)!))))/(2*n-1)). - Vladimir Kruchinin, May 16 2013

More terms from Vladeta Jovovic, Aug 11 2002

Edited by Johannes W. Meijer, Sep 21 2012

A002198 Denominators of coefficients for numerical integration.

Original entry on oeis.org

24, 5760, 967680, 464486400, 122624409600, 2678117105664000, 64274810535936000, 149852129706639360000, 669659197233029971968000, 8839501403475995629977600000, 4879404774718749587747635200000

Offset: 0

N. J. A. Sloane

nonn

The denominators of these coefficients for numerical integration are a combination of the Bernoulli numbers B{2k}, the central factorial numbers A008956(n, k) and the factor 4^n*(2*n+1)!. - Johannes W. Meijer, Jan 27 2009

H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218.
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).

T. D. Noe, Table of n, a(n) for n = 0..100
H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218. [Annotated scanned copy]
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 545.

Cf. A002197.
See A000367, A006954, A008956 and A002671 for underlying sequences.
Factor of the LS1[ -2,n] matrix coefficients in A160487.

nmax:=10: for n from 0 to nmax do A008956(n, 0) := 1: A008956(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax do for k from 1 to n-1 do A008956(n, k) := (2*n-1)^2*A008956(n-1, k-1) + A008956(n-1, k) od: od:
for n from 0 to nmax do Delta(n) := add((1-2^(2*k1-1)) * (-1)^k1 * (bernoulli(2*k1)/(2*k1)) * A008956(n, n+1-k1), k1=1..n+1) / (2*4^(n)*(2*n+1)!) end do: a:=n-> denom (Delta(n)): seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 27 2009, Revised Sep 21 2012

a(n) = denominator(Sum_{k=1..n+1}((1-2^(2*k-1))*(-1)^k*(B_{2k}/(2*k))*A008956(n, n+1-k)) / (2*4^(n)*(2*n+1)!)) for n >= 0. - Johannes W. Meijer, Jan 27 2009

Two more terms and editing by Johannes W. Meijer, Sep 21 2012

A002453 Central factorial numbers: 2nd subdiagonal of A008958.

Original entry on oeis.org

1, 35, 966, 24970, 631631, 15857205, 397027996, 9931080740, 248325446061, 6208571999575, 155218222621826, 3880490869237710, 97012589464171291, 2425317596203339145, 60632965641474990456, 1515824372664398367880

Offset: 0

N. J. A. Sloane, Simon Plouffe

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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).
T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 36.

G. C. Greubel, Table of n, a(n) for n = 0..710
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to factorial numbers
Index entries for linear recurrences with constant coefficients, signature (35,-259,225).

Right-hand column 2 in triangle A008958.

Cf. A002452.

List([0..20],n->(5^(2*n+4)-3^(2*n+5)+2)/384); # Muniru A Asiru, Dec 20 2018

[(5^(2*n+4)-3^(2*n+5)+2)/384: n in [0..20]]; // G. C. Greubel, Jul 04 2019

A002453:=-1/(z-1)/(25*z-1)/(9*z-1); # Simon Plouffe (from his 1992 dissertation).

CoefficientList[Series[1/((1-x)(1-9x)(1-25x)),{x,0,20}],x] (* or *) LinearRecurrence[{35,-259,225},{1,35,966},20] (* Harvey P. Dale, Feb 25 2015 *)

vector(20, n, n--; (5^(2*n+4)-3^(2*n+5)+2)/384) \\ G. C. Greubel, Jul 04 2019

[(5^(2*n+4)-3^(2*n+5)+2)/384 for n in (0..20)] # G. C. Greubel, Jul 04 2019

G.f.: 1/((1 - x)*(1 - 9*x)*(1 - 25*x)).
a(n) = (5^(2*n + 4) - 3^(2*n + 5) + 2)/384.
E.g.f.: sinh(x)^5/120 = Sum_{n>=0} a(n)*x^(2*n + 5)/(2*n + 5)!. - Vladimir Kruchinin, Sep 30 2012
a(n) = det(|v(i+3,j+2)|, 1 <= i,j <= n), where v(n,k) are central factorial numbers of the first kind with odd indices (A008956). - Mircea Merca, Apr 06 2013
a(n) = 35*a(n-1) -259*a(n-2) +225*a(n-3), with a(0) = 1, a(1) = 35, a(2) = 966. - Harvey P. Dale, Feb 25 2015
a(n) = 25*a(n-1) + A002452(n+1), with a(0) = 1. - Nadia Lafreniere, Aug 08 2022

A182867 Triangle read by rows: row n gives coefficients in expansion of Product_{i=1..n} (x - (2i)^2), highest powers first.

Original entry on oeis.org

1, 1, -4, 1, -20, 64, 1, -56, 784, -2304, 1, -120, 4368, -52480, 147456, 1, -220, 16368, -489280, 5395456, -14745600, 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400, 1, -560, 119392, -12263680, 633721088, -15658639360, 157294854144, -416179814400, 1, -816, 262752, -42828032, 3773223168, -177891237888, 4165906530304, -40683662475264, 106542032486400, 1, -1140, 527136, -127959680, 17649505536, -1400415544320, 61802667606016, -1390437378293760, 13288048674471936, -34519618525593600

Offset: 0

N. J. A. Sloane, Feb 01 2011

These are scaled central factorial numbers (see the discussion in the Comments section of A008955). The coefficients in the expansion of Product_{i=1..n} (x - i^2) give A008955, and the coefficients in the expansion of Product_{i=1..n} (x - (2i+1)^2) give A008956.

			Triangle begins:
 1
 1, -4
 1, -20, 64
 1, -56, 784, -2304
 1, -120, 4368, -52480, 147456
 1, -220, 16368, -489280, 5395456, -14745600
 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400
 1, -560, 119392, -12263680, 633721088, -15658639360, 157294854144, -416179814400
 1, -816, 262752, -42828032, 3773223168, -177891237888, 4165906530304, -40683662475264, 106542032486400
 1, -1140, 527136, -127959680, 17649505536, -1400415544320, 61802667606016, -1390437378293760, 13288048674471936, -34519618525593600
...
For example, for n=2, (x-4)(x-16) = x^2 - 20x + 64 => [1, -20, 64].

T. L. Curtright, D. B. Fairlie, and C. K. Zachos, A compact formula for rotations as spin matrix polynomials, arXiv preprint arXiv:1402.3541 [math-ph], 2014.
T. L. Curtright and T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arXiv:1408.0767 [math-ph], 2014.
T. L. Curtright, More on Rotations as Spin Matrix Polynomials, arXiv preprint arXiv:1506.04648 [math-ph], 2015.

Cf. A008955, A008956. This triangle is formed from the even-indexed rows of A182971 (the odd-indexed rows give A008956).

Cf. A160563.

Q:= n -> if n mod 2 = 0 then sort(expand(mul(x-4*i^2,i=1..n/2)));
else sort(expand(mul(x-(2*i+1)^2,i=0..(n-1)/2))); fi;
for n from 0 to 10 do
t1:=eval(Q(2*n)); t1d:=degree(t1);
t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
lprint(t2);
od:
# Using a bivariate generating function (adding a superdiagonal 1,0,0, ...):
gf := (t + sqrt(1 + t^2))^x:
ser := series(gf, t, 20): ct := n -> coeff(ser, t, n):
T := (n, k) -> n!*coeff(ct(n), x, n - k):
EvenPart := (T, len) -> local n, k;
seq(print(seq(T(n, k), k = 0..n, 2)), n = 0..2*len-1, 2):
EvenPart(T, 6);  # Peter Luschny, Mar 03 2024

Given a (0, 0)-based triangle U we call the triangle [U(n, k), k=0..n step 2, n=0..len step 2] the 'even subtriangle' of U. This triangle is the even subtriangle of U(n, k) = n! * [x^(n-k)] [t^n] (t + sqrt(1 + t^2))^x, albeit adding a superdiagonal 1, 0, 0, ... See A160563 for the odd subtriangle. - Peter Luschny, Mar 03 2024

A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)(x-(n-2)^2)(x-(n-4)^2)...(x-(1 or 2)^2), highest powers first.

Original entry on oeis.org

1, 1, -1, 1, -4, 1, -10, 9, 1, -20, 64, 1, -35, 259, -225, 1, -56, 784, -2304, 1, -84, 1974, -12916, 11025, 1, -120, 4368, -52480, 147456, 1, -165, 8778, -172810, 1057221, -893025, 1, -220, 16368, -489280, 5395456, -14745600, 1, -286, 28743, -1234948, 21967231, -128816766, 108056025, 1, -364, 48048, -2846272, 75851776, -791691264, 2123366400

Offset: 0

N. J. A. Sloane, Feb 01 2011

These are scaled versions of the central factorial numbers in A008955 and A008956.
Even-indexed rows give A182867, odd-indexed rows give A008956.
A121408 is an unsigned and aerated version of the row reverse of this triangle. - Peter Bala, Aug 29 2012

			Triangle begins:
1
1, -1
1, -4
1, -10, 9
1, -20, 64
1, -35, 259, -225
1, -56, 784, -2304
1, -84, 1974, -12916, 11025
1, -120, 4368, -52480, 147456
1, -165, 8778, -172810, 1057221, -893025
1, -220, 16368, -489280, 5395456, -14745600
...
E.g. for n=5 Q(5) = (x-1^2)*(x-3^2)*(x-5^2) = x^3-35*x^2+259*x-225.

Even-indexed rows give A182867, odd-indexed rows give A008956.
Column 1,4,10,20, ... is A000292. The next two columns give A181888, A184878. The last diagonal is A184877.
Cf. A008955, A008956. A121408.

Q:= n -> if n mod 2 = 0 then sort(expand(mul(x-4*i^2,i=1..n/2)));
else sort(expand(mul(x-(2*i+1)^2,i=0..(n-1)/2))); fi;
for n from 0 to 12 do
t1:=eval(Q(n)); t1d:=degree(t1);
t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
lprint(t2);
od:

For n even, let Q(n) = Product_{i=1..n/2} (x - (2*i)^2) and for n odd let Q(n) = Product_{i=0..(n-1)/2} (x - (2i+1)^2). n-th row of triangle gives coefficients in expansion of Q(n).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A160476 The first right hand column of the Zeta and Lambda triangles.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula

A161736 Denominators of the column sums of the BG2 matrix.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A162448 Numerators of the column sums of the LG1 matrix.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Formula

Extensions

A002197 Numerators of coefficients for numerical integration.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

Extensions

A002198 Denominators of coefficients for numerical integration.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

A182971 Triangle read by rows: coefficients in expansion of Q(n) = (x-n^2)(x-(n-2)^2)(x-(n-4)^2)...(x-(1 or 2)^2), highest powers first.