0,5

The terms in row n are the coefficients of Prod[i=1..n, x+i^2].

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 2009: (Start)

We define Beta(n-z,n+z)/Beta(n,n) = Gamma(n-z)*Gamma(n+z)/Gamma(n)^2 = sum(EG2[2m,n]*z^(2m), m = 0..infinity) with Beta(z,w) the Beta function. The EG2[2m,n] coefficients are quite interesting, see A161739. Our definition leads to EG2[2m,1] = 2*eta(2m) and the recurrence relation EG2[2m,n] = EG2[2m,n-1] - EG2[2m-2,n-1]/(n-1)^2 for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. , with eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function. We found for the matrix coefficients 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 the central factorial numbers t1(n,m) as defined above, see also the Maple program.

From the EG2 matrix we arrive at the ZG2 matrix, see A161739 for its odd counterpart, which is defined by ZG2[2m,1] = 2*zeta(2m) and the recurrence relation ZG2(2m,n) = ZG2(2m-2,n-1)/(n*(n-1))-(n-1)*ZG2(2m,n-1)/n for m = -2, -1, 0, 1, 2, .. and n = 2, 3, .. . We found for the ZG2[2m,n] = sum((-1)^(k+1)*t1(n-1,k-1)*2*zeta(2*m-2*n+2*k)/((n-1)!*(n)!), k=1..n) and we see that the central factorial numbers t1(n,m) once again play a crucial role.

(End)

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

P.L. Butzer, M. Schmidt, E.L. Stark, L. Vogt, Central Factorial Numbers: Their main properties and some applications, Numerical Functional Analysis and Optimization, 10 (5&6), 419-488 (1989). [From Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 2009]

T. D. Noe, Rows n=0..50 of triangle, flattened

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. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 2009]

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. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 2009]

t1(n,m) = t1(n-1,m) + n^2*t1(n-1,m-1) with t1(n,0) = 1 and t1(n,n) = (n!)^2.

1; 1,1; 1,5,4; 1,14,49,36; 1,30,273,820,576; ...

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 2009: (Start)

nmax:=8: mmax:=nmax: for n from 0 to nmax do t1(n, 0):=1 end do: for n from 0 to nmax do t1(n, n):=(n!)^2 end do: for n from 1 to nmax 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: T:=0: for n from 0 to nmax do for m from 0 to n do a(T):=t1(n, m): T:=T+1 od: od: seq(a(n), n=0..T-1);

(End)

Cf. A036969.

Columns include A000330, A000596, A000597. Right-hand columns include A001044, A001819, A001820, A001821. Row sums are in A101686.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 2009: (Start)

Appears in A160464 (Eta triangle), A160474 (Zeta triangle), A160479 (ZL(n) sequence), A161739 (RSEG2 triangle), A161742, A161743, A002195 and A002196.

(End)

Sequence in context: A072222 A005752 A098494 this_sequence A152862 A108440 A102220

tabl,nonn,nice,easy,new

N. J. A. Sloane (njas(AT)research.att.com).

There's an error in the last column of Riordan's table (change 46076 to 21076).

More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 16 2000