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A162440
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The pg(n) sequence that is associated with the Eta triangle A160464.
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8
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2, 16, 144, 4608, 115200, 4147200, 203212800, 26011238400, 2106910310400, 210691031040000, 25493614755840000, 3671080524840960000, 620412608698122240000, 121600871304831959040000
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OFFSET
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2,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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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.
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FORMULA
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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].
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EXAMPLE
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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.
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MAPLE
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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;
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CROSSREFS
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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 r(n) sequence equals A062383(n) (n>=1).
The e(n) sequence equals A029837(n) (n>=1).
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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