0,9

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.

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, .. .

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.

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.

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).

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 22 2009: (Start)

The a(n) formulae of the right hand columns are related to sequence A036283, see also A161740 and A161741.

(End)

J.W. Meijer and N.H.G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.

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.

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, .. .

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)

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, .. .

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.

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.

The first few expressions for the ZG1[2*m-1,p+1] coefficients are:

ZG1[2*m-1,1] = (zeta(2*m-1))/(1/2)

ZG1[2*m-1,2] = (zeta(2*m-3) - zeta(2*m-1))/1

ZG1[2*m-1,3] = (zeta(2*m-5) - 5*zeta(2*m-3)+4*zeta(2*m-1))/6

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

The first few rs(2*m,p) are (m >= p+2)

rs(2*m,p=0) = ZG1[2*m-1,1]

rs(2*m, p=1) = ZG1[2*m-1,1] + ZG1[2*m-1,2]

rs(2*m, p=2) = ZG1[2*m-1,1] + 3*ZG1[2*m-1,2] + 2*ZG1[2*m-1,3]

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]

The first few rs(2*m,p) are (m >= p+2)

rs(2*m, p=-1) = zeta(2*m+1)/(1/2)

rs(2*m, p=0) = zeta(2*m-1)/(1/2)

rs(2*m, p=1) = (zeta(2*m-1)+zeta(2*m-3))/1

rs(2*m, p=2) = (zeta(2*m-1)+4*zeta(2*m-3)+zeta(2*m-5))/3

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

The first few rows of the RSEG2 triangle are:

[1]

[0, 1]

[0, 1, 1]

[0, 1, 4, 1]

[0, 0, 13, 10, 1]

[0, -4, 30, 73, 20, 1]

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;

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;

A000007, A129825, A161742 and A161743 are the first four left hand columns.

A000012, A000292, A107963, A161740 and A161741 are the first five right hand columns.

A010790 equals 2*r(n) and A054977 equals denom(r(n)).

A001710 equals numer(q(n)) and A141044 equals denom(q(n)).

A000142 equals the row sums.

A008955 is a central factorial number triangle.

A028246 is Worpitzky's triangle.

Sequence in context: A049763 A085992 A117411 this_sequence A094924 A056968 A035253

Adjacent sequences: A161736 A161737 A161738 this_sequence A161740 A161741 A161742

easy,sign,tabl

Johannes W. Meijer & Nico Baken (meijgia(AT)hotmail.com and n.h.g.baken(AT)tudelft.nl), Jun 18 2009