Search: id:A008955
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%I A008955
%S A008955 1,1,1,1,5,4,1,14,49,36,1,30,273,820,576,1,55,1023,7645,21076,14400,
%T A008955 1,91,3003,44473,296296,773136,518400,1,140,7462,191620,2475473,
%U A008955 15291640,38402064,25401600,1,204,16422,669188,14739153,173721912
%N A008955 Triangle of central factorial numbers t(2n,2n-2k) read by rows.
%C A008955 The terms in row n are the coefficients of Prod[i=1..n, x+i^2].
%C A008955 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 
               2009: (Start)
%C A008955 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.
%C A008955 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.
%C A008955 (End)
%D A008955 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.
%D A008955 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A008955 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]
%H A008955 T. D. Noe, <a href="b008955.txt">Rows n=0..50 of triangle, flattened</
               a>
%H A008955 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. [From Johannes W. 
               Meijer (meijgia(AT)hotmail.com), Jun 18 2009]
%H A008955 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. [From Johannes W. Meijer (meijgia(AT)hotmail.com), 
               Jun 18 2009]
%F A008955 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.
%e A008955 1; 1,1; 1,5,4; 1,14,49,36; 1,30,273,820,576; ...
%p A008955 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 
               2009: (Start)
%p A008955 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);
%p A008955 (End)
%Y A008955 Cf. A036969.
%Y A008955 Columns include A000330, A000596, A000597. Right-hand columns include 
               A001044, A001819, A001820, A001821. Row sums are in A101686.
%Y A008955 Appears in A160464 (Eta triangle), A160474 (Zeta triangle), A160479 (ZL(n)), 
               A161739 (RSEG2 triangle), A161742, A161743, A002195 , A002196 , A162440 
               (EG1 matrix), A162446 (ZG1 matrix) and A163927. [From Johannes W. 
               Meijer (meijgia(AT)hotmail.com), Jun 18 2009, Jul 06 2009 and Aug 
               17 2009.]
%Y A008955 Sequence in context: A072222 A005752 A098494 this_sequence A152862 A108440 
               A102220
%Y A008955 Adjacent sequences: A008952 A008953 A008954 this_sequence A008956 A008957 
               A008958
%K A008955 tabl,nonn,nice,easy
%O A008955 0,5
%A A008955 N. J. A. Sloane (njas(AT)research.att.com).
%E A008955 There's an error in the last column of Riordan's table (change 46076 
               to 21076).
%E A008955 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 16 2000
%E A008955 Link added and cross-references edited by Johannes W. Meijer (meijgia(AT)hotmail.com), 
               Aug 17 2009

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