A161739 - OEIS

%I #26 Mar 24 2022 10:33:18

%S 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,

%T 0,120,-504,1561,3008,798,56,1,0,0,736,-2856,25809,14572,1974,84,1,0,

%U -12096,44640,-73520,125580,218769,55060,4326,120,1

%N The RSEG2 triangle.

%C The EG2[2*m,n] matrix coefficients were introduced in A008955. We discovered that EG2[2m,n] = Sum_{k = 1..n} (-1)^(k+n)*t1(n-1,k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2 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 A different way to define these matrix coefficients is EG2[2*m,n] = (1/m)*Sum_{k = 0..m-1} ZETA(2*m-2*k, n-1)*EG2[2*k, n] with ZETA(2*m, n-1) = zeta(2*m) - Sum_{k = 1..n-1} (k)^(-2*m) and EG2[0, n] = 1, for m = 0, 1, 2, ..., and n = 1, 2, 3, ... .

%C We define the row sums of the EG2 matrix rs(2*m,p) = Sum_{n >= 1} (n^p)*EG2(2*m,n) 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) formulas, see the examples.

%C 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 formulas.

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

%H 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.

%H J. W. Meijer and N.H.G. Baken, <a href="https://doi.org/10.1016/0167-7152(87)90041-1">The Exponential Integral Distribution</a>, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.

%F RGEG2(2*m,z) = Sum_{n >= 1} EG2[2*m,n]*z^(n-1) = Integral_{y = 0..oo}((2*y)^(2*m)/(2*m)!)* cosh(y)/(cosh(y)^2 - z)^(3/2) for m >= 0.

%F EG2[2*m,n] = Sum_{k = 1..n} (-1)^(k+n)* A008955(n-1, k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2.

%F ZG1[2*m-1,p+1] = Sum_{j = 0..p} (-1)^j*A008955(p, j)*zeta(2*m-(2*p+1-2*j))/ r(p) with r(p)= p!*(p+1)!/2 and p >= 0.

%F rs(2*m,p) = Sum_{k = 0..p} A028246(p+1,k+1)*ZG1[2*m-1,k+1] and p >= 0; p <= m-2.

%F rs(2*m,p) = Sum_{k = 0..p+1} A161739(p+1,k)*zeta(2*m+1-2*k)/q(p+1) with q(p+1) = (p+1)!/2 and p >= -1; p <= m-2.

%F From _Peter Bala_, Mar 19 2022: (Start)

%F It appears that the k-th row polynomial (with indexing starting at k = 1) is given by R(k,n^2) = (k-1)!*Sum_{i = 0..n} (-1)^(n-i)*(i^k)* binomial(n,i)*binomial(n+i,i)/(n+i) for n >= 1.

%F For example, for k = 6, Maple's SumTools:-Summation procedure gives 5!*Sum_{i = 0..n} (-1)^(n-i)*(i^6)*binomial(n,i)*binomial(n+i,i)/(n+i) = -4*n^2 + 30*n^4 + 73*n^6 + 20*n^8 + n^10 = R(6,n^2). (End)

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

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

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

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

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

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

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

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

%e   rs(2*m, p=3) = ZG1[2*m-1,1] + 7*ZG1[2*m-1,2] + 12*ZG1[2*m-1,3] + 6*ZG1[2*m-1,4]

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

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

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

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

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

%e   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 The first few rows of the RSEG2 triangle are:

%e   [1]

%e   [0, 1]

%e   [0, 1, 1]

%e   [0, 1, 4, 1]

%e   [0, 0, 13, 10, 1]

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

%p 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, k1+1)*(sum((-1)^(j)*A008955(k1, j)*2*x^(2*nmax-(2*k1+1-2*j)), j=0..k1)/ (k1!*(k1+1)!)), k1=0..i) od: a(0,0) := 1: for n from 1 to nmax-1 do for m from 0 to n do a(n,m) := coeff(s(n-1), x, 2*nmax-1-2*m+2) od: od: seq(seq(a(n, m), m=0..n), n=0..nmax-1); for n from 0 to nmax-1 do seq(a(n, m), m=0..n) od;

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

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

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

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

%Y A000142 equals the row sums.

%Y A008955 is a central factorial number triangle.

%Y A028246 is Worpitzky's triangle.

%K easy,sign,tabl

%O 0,9

%A _Johannes W. Meijer_ & Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

%E Minor error corrected and edited by _Johannes W. Meijer_, Sep 22 2012