A160487 -id:A160487 - cp's OEIS frontend

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

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]

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]

A036282 Write cosec x = 1/x + Sum_{n>=1} e_n * x^(2n-1)/(2n-1)!; sequence gives numerators of e_n.

Original entry on oeis.org

1, 7, 31, 127, 511, 1414477, 8191, 118518239, 5749691557, 91546277357, 162912981133, 1982765468311237, 22076500342261, 455371239541065869, 925118910976041358111, 16555640865486520478399, 1302480594081611886641, 904185845619475242495834469891

Offset: 1

N. J. A. Sloane

From Johannes W. Meijer, May 24 2009: (Start)
Absolute value of numerator of [2^(2n-1) - 1] * Bernoulli(2n)/n.
Equals the absolute values of the numerators of the LS1[ -2*m,n=1] matrix coefficients of A160487 for m = 1, 2, .. ,.
(End)

			cosec x
= x^(-1) + 1/6*x + 7/360*x^3 + 31/15120*x^5 + ...
= x^(-1) + 1/6 * x/1! + 7/60 * x^3/3! + 31/126 * x^5/5! + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).

Seiichi Manyama, Table of n, a(n) for n = 1..275
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.68).
R. P. Brent, Asymptotic approximation of central binomial coefficients with rigorous error bounds, arXiv:1608.04834 [math.NA], 2016.
Duane W. DeTemple, Shun-Hwa Wang, Half-integer approximations for the partial sums of harmonic series, J. Math. Anal. Applic. 160 (1991) 149-156
Simon Plouffe, On the values of the functions zeta and gamma, arXiv:1310.7195 [math.NT], 2013.
Eric Weisstein's World of Mathematics, Cosecant
Eric Weisstein's World of Mathematics, Riemann-Siegel Function
Wikipedia, Trigonometric functions

Cf. A036280, A036281, A036283.
Cf. A160487.
Differs from A282898.

a:= n-> (m-> numer(coeff(series(csc(x), x, m+1), x, m)*m!))(2*n-1):
seq(a(n), n=1..20);  # Alois P. Heinz, Jun 21 2018

a[n_] := Abs[ Numerator[ (2^(2*n-1)-1) * BernoulliB[2*n]/n ] ]; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, May 31 2013, after Johannes W. Meijer *)

a(n) = abs(numerator((2^(2*n-1)-1)*bernfrac(2*n)/n)); \\ Michel Marcus, Mar 01 2015

Title corrected and offset changed by Johannes W. Meijer, May 21 2009

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

cp's OEIS Frontend

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

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A162448 Numerators of the column sums of the LG1 matrix.

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A036282 Write cosec x = 1/x + Sum_{n>=1} e_n * x^(2n-1)/(2n-1)!; sequence gives numerators of e_n.

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A002197 Numerators of coefficients for numerical integration.

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A002198 Denominators of coefficients for numerical integration.

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