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%I A161739
%S A161739 1,0,1,0,1,1,0,1,4,1,0,0,13,10,1,0,4,30,73,20,1,0,0,14,425,273,35,1,0,
%T A161739 120,504,1561,3008,798,56,1,0,0,736,2856,25809,14572,1974,84,1,0,
%U A161739 12096,44640,73520,125580,218769,55060,4326,120,1
%V A161739 1,0,1,0,1,1,0,1,4,1,0,0,13,10,1,0,-4,30,73,20,1,0,0,-14,425,273,35,1,
               0,
%W A161739 120,-504,1561,3008,798,56,1,0,0,736,-2856,25809,14572,1974,84,1,0,
%X A161739 -12096,44640,-73520,125580,218769,55060,4326,120,1
%N A161739 The RSEG2 triangle
%C A161739 The EG2[2*m,n] matrix coefficients were introduced in A008955. We discovered 
               that EG2[2m,n] = sum((-1)^(k+n)*t1(n-1,k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2, 
               k=1..n) with t1(n,m) the central factorial numbers A008955 and eta(m) 
               = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and 
               zeta(m) the Riemann zeta function.
%C A161739 A different way to define these matrix coefficients is EG2[2*m,n] = (1/
               m)*sum(ZETA(2*m-2*k, n-1)*EG2[2*k, n],k=0..m-1) with ZETA(2*m, n-1) 
               = zeta(2*m)-sum((k)^(-2*m), k=1..n-1) and EG2[0, n] = 1, for m = 
               0, 1, 2, .. , and n = 1, 2, 3, .. .
%C A161739 We define the row sums of the EG2 matrix rs(2*m,p) = sum((n^p)*EG2(2*m,
               n), n = 1..infinity) for p = -2, -1, 0, 1, .. and m >= p+2. We discovered 
               that rs(2*m,p=-2) = 2*eta(2*m+2) = (1-2^(1-(2*m+2)))*zeta(2*m+2). 
               This formula is quite unlike the other rs(2*m,p) formulae, see the 
               examples.
%C A161739 The series expansions of the row generators RGEG2(z,2*m) about z= 0 lead 
               to the EG2[2*m,n] coefficients while the series expansions about 
               z=1 lead to the ZG1[2*m-1,n] coefficients, see the formulae.
%C A161739 The first Maple program gives the triangle coefficients. Adding the second 
               program to the first one gives information about the row sums rs(2*m,
               p).
%C A161739 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 22 
               2009: (Start)
%C A161739 The a(n) formulae of the right hand columns are related to sequence A036283, 
               see also A161740 and A161741.
%C A161739 (End)
%H A161739 J.W. Meijer and N.H.G. Baken, <a href="http://www.sciencedirect.com/science/
               article/B6V1D-47N62TS-4W/2/faab5428aaed95db7cfb6e62c3c93e97">The 
               Exponential Integral Distribution</a>, Statistics and Probability 
               Letters, Volume 5, No.3, April 1987. pp 209-211.
%H A161739 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, 
               Chapter 23, pp. 811-812.
%F A161739 RGEG2(2*m,z) = sum(EG2[2*m,n]*z^(n-1), n=1..infinity) = int(((2*y)^(2*m)/
               (2*m)!)* cosh(y)/(cosh(y)^2-z)^(3/2), y = 0..infinity) for m = 0, 
               1, 2, .. .
%F A161739 EG2[2m,n] = sum((-1)^(k+n)* A008955(n-1, k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2, 
               k=1..n)
%F A161739 ZG1[2*m-1,p+1] = sum((-1)^j*A008955(p, j)*zeta(2*m-(2*p+1-2*j)), j = 
               0..p)/r(p) with r(p)= p!*(p+1)!/2 and p = 0, 1, 2, .. .
%F A161739 rs(2*m,p) = sum(A028246(p+1,k+1)*ZG1[2*m-1,k+1], k = 0..p) and p = 0, 
               1, .. ; p =< m-2.
%F A161739 rs(2*m,p) = sum(A161739(p+1,k)*zeta(2*m+1-2*k), k = 0..p+1)/q(p+1) with 
               q(p+1) = (p+1)!/2 and p = -1, 0, 1, 2, .. ; p =< m-2.
%e A161739 The first few expressions for the ZG1[2*m-1,p+1] coefficients are:
%e A161739 ZG1[2*m-1,1] = (zeta(2*m-1))/(1/2)
%e A161739 ZG1[2*m-1,2] = (zeta(2*m-3) - zeta(2*m-1))/1
%e A161739 ZG1[2*m-1,3] = (zeta(2*m-5) - 5*zeta(2*m-3)+4*zeta(2*m-1))/6
%e A161739 ZG1[2*m-1,4] = (zeta(2*m-7) - 14*zeta(2*m-5)+49*zeta(2*m-3)-36*zeta(2*m-1))/
               72
%e A161739 The first few rs(2*m,p) are (m >= p+2)
%e A161739 rs(2*m,p=0) = ZG1[2*m-1,1]
%e A161739 rs(2*m, p=1) = ZG1[2*m-1,1] + ZG1[2*m-1,2]
%e A161739 rs(2*m, p=2) = ZG1[2*m-1,1] + 3*ZG1[2*m-1,2] + 2*ZG1[2*m-1,3]
%e A161739 rs(2*m, p=3) = ZG1[2*m-1,1] + 7*ZG1[2*m-1,2] + 12*ZG1[2*m-1,3] + ZG1[2*m-1,
               4]
%e A161739 The first few rs(2*m,p) are (m >= p+2)
%e A161739 rs(2*m, p=-1) = zeta(2*m+1)/(1/2)
%e A161739 rs(2*m, p=0) = zeta(2*m-1)/(1/2)
%e A161739 rs(2*m, p=1) = (zeta(2*m-1)+zeta(2*m-3))/1
%e A161739 rs(2*m, p=2) = (zeta(2*m-1)+4*zeta(2*m-3)+zeta(2*m-5))/3
%e A161739 rs(2*m, p=3) = (0*zeta(2*m-1)+13*zeta(2*m-3)+10*zeta(2*m-5)+zeta(2*m-7))/
               12
%e A161739 The first few rows of the RSEG2 triangle are:
%e A161739 [1]
%e A161739 [0, 1]
%e A161739 [0, 1, 1]
%e A161739 [0, 1, 4, 1]
%e A161739 [0, 0, 13, 10, 1]
%e A161739 [0, -4, 30, 73, 20, 1]
%p A161739 restart; nmax:=10; for n from 0 to nmax do A008955(n,0):=1 end do: for 
               n from 0 to nmax do A008955(n,n):=(n!)^2 end do: for n from 1 to 
               nmax do for m from 1 to n-1 do A008955(n,m):= A008955(n-1,m-1)*n^2+A008955(n-1,
               m) end do: end do: for n from 1 to nmax do A028246(n,1):=1 od: for 
               n from 1 to nmax do A028246(n,n):=(n-1)! od: for n from 3 to nmax 
               do for m from 2 to n-1 do A028246(n,m):=m*A028246(n-1,m)+(m-1)*A028246(n-1,
               m-1) od: od: for i from 0 to nmax-2 do s(i):=((i+1)!/2)*sum(A028246(i+1,
               k+1)*(sum((-1)^(j)* A008955(k,j)*2*x^(2*nmax-(2*k+1-2*j)), j=0..k)/ 
               (k!*(k+1)!)),k=0..i) od: a(0,0):=1: ax(0):= a(0,0): T:=1: for n from 
               1 to nmax-1 do a(n,0):=0: ax(T):=a(n,0): T:=T+1; for m from 1 to 
               n do a(n,m):=coeff(s(n-1),x,2*nmax-1-2*m+2): ax(T):=a(n,m): T:=T+1 
               od: od: seq(ax(n),n=0..T-1); for n from 0 to nmax-1 do seq(a(n,m),
               m=0..n) od;
%p A161739 m:=7: row:=2*m; rs(2*m,-2):= 2*eta(2*m+2); for p from -1 to m-2 do q(p+1) 
               := (p+1)!/2 od: for p from -1 to m-2 do rs(2*m,p) := sum(a(p+1,k)*zeta(2*m+1-2*k),
               k=0..p+1)/q(p+1) od;
%Y A161739 A000007, A129825, A161742 and A161743 are the first four left hand columns.
%Y A161739 A000012, A000292, A107963, A161740 and A161741 are the first five right 
               hand columns.
%Y A161739 A010790 equals 2*r(n) and A054977 equals denom(r(n)).
%Y A161739 A001710 equals numer(q(n)) and A141044 equals denom(q(n)).
%Y A161739 A000142 equals the row sums.
%Y A161739 A008955 is a central factorial number triangle.
%Y A161739 A028246 is Worpitzky's triangle.
%Y A161739 Sequence in context: A049763 A085992 A117411 this_sequence A094924 A056968 
               A035253
%Y A161739 Adjacent sequences: A161736 A161737 A161738 this_sequence A161740 A161741 
               A161742
%K A161739 easy,sign,tabl
%O A161739 0,9
%A A161739 Johannes W. Meijer & Nico Baken (meijgia(AT)hotmail.com and n.h.g.baken(AT)tudelft.nl), 
               Jun 18 2009

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