A160474 -id:A160474 - 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]

A162446 Numerators of the column sums of the ZG1 matrix.

Original entry on oeis.org

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

Offset: 2

Johannes W. Meijer, Jul 06 2009

The ZG1 matrix coefficients are defined by ZG1[2m-1,1] = 2*zeta(2m-1) for m = 2, 3, .. , and the recurrence relation ZG1[2m-1,n] = (ZG1[2m-3,n-1] - (n-1)^2*ZG1[2m-1,n-1])/(n*(n-1)) with m = .. , -2, -1, 0, 1, 2, .. and n = 1, 2, 3, .. , under the condition that n <= (m-1). As usual zeta(m) is the Riemann zeta function. For the ZG2 matrix, the even counterpart of the ZG1 matrix, see A008955.
These two formulas enable us to determine the values of the ZG1[2*m-1,n] coefficients, with m all integers and n all positive integers, but not for all. If we choose, somewhat but not entirely arbitrarily, ZG1[1,1] = 2*gamma, with gamma the Euler-Mascheroni constant, we can determine them all.
The coefficients in the columns of the ZG1 matrix, for m >= 1 and n >= 2, can be generated with GFZ(z;n) = (hg(n)*CFN1(z;n)*GFZ(z;n=1) + ZETA(z;n))/pg(n) with pg(n) = 6*(n-1)!* (n)!*A160476(n) and hg(n) = 6*A160476(n). For the CFN1(z;n) and the ZETA(z;n) polynomials see A160474.
The column sums cs(n) = sum(ZG1[2*m-1,n], m = 1 .. infinity), for n >= 2, of the ZG1 matrix can be determined with the first Maple program. In this program we have made use of the remarkable fact that if we take ZGx[2*m-1,n] = 2, for m >= 1, and ZGx[ -1,n] = ZG1[ -1,n] and assume that the recurrence relation remains the same we find that the column sums of this new matrix converge to the same values as the original cs(n).
The ZG1[2*m-1,n] matrix coefficients can be generated with the second Maple program.
The ZG1 matrix is related to the ZS1 matrix, see A160474 and the formulas below.

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

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.

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

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

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

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

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