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

%I A162440 #20 May 08 2018 15:11:56
%S A162440 2,16,144,4608,115200,4147200,203212800,26011238400,2106910310400,
%T A162440 210691031040000,25493614755840000,3671080524840960000,
%U A162440 620412608698122240000,121600871304831959040000
%N A162440 The pg(n) sequence that is associated with the Eta triangle A160464.
%C A162440 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.
%C A162440 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.
%C A162440 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.
%C A162440 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.
%C A162440 The EG1 matrix is related to the ES1 matrix, see A160464 and the formulas below.
%D A162440 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.
%H A162440 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.
%F A162440 pg(n) = (n-1)!^2*2^floor(log(n-1)/log(2)+1) for n >= 2.
%F A162440 r(n) = 2^e(n) = 2^floor(log(n-1)/log(2)+1) for n >= 2.
%F A162440 EG1[ -1,n] = 2^(1-2*n)*(2*n-1)!/((n-1)!^2) for n >= 1.
%F A162440 GFE(z;n) = sum (EG1[2*m-1,n]*z^(2*m-2), m=1..infinity).
%F A162440 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.
%F A162440 EG1[2m-1,n] = (2*2^(1-2*n)*(2*n-1)!/((n-1)!^2)) * ES1[2m-1,n].
%e A162440 The first few generating functions GFE(z;n) are:
%e A162440 GFE(z;n=2) = ((-1)*2*(z^2 - 1)*GFE(z;n=1) + (-1))/2,
%e A162440 GFE(z;n=3) = ((+1)*4*(z^4 - 5*z^2 + 4)*GFE(z;n=1) + (-11 + 2*z^2))/16,
%e A162440 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.
%p A162440 nmax := 16; seq((n-1)!^2*2^floor(ln(n-1)/ln(2)+1), n=2..nmax);
%p A162440 # End program 1
%p A162440 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;
%p A162440 # End program 2 (Edited by _Johannes W. Meijer_, Sep 21 2012)
%Y A162440 The ETA(z, n) polynomials and the ES1 matrix lead to the Eta triangle A160464.
%Y A162440 The CFN1(z, n), the t1(n, m) and the EG2 matrix lead to A008955.
%Y A162440 The EG1[ -1, n] equal (1/2)*A001803(n-1)/A046161(n-1).
%Y A162440 The r(n) sequence equals A062383(n) (n>=1).
%Y A162440 The e(n) sequence equals A029837(n) (n>=1).
%Y A162440 Cf. A160473 (p(n) sequence).
%Y A162440 Cf. A162443 (BG1 matrix), A162446 (ZG1 matrix) and A162448 (LG1 matrix).
%K A162440 easy,nonn
%O A162440 2,1
%A A162440 _Johannes W. Meijer_, Jul 06 2009