Search: id:A008956
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%I A008956
%S A008956 1,1,1,1,10,9,1,35,259,225,1,84,1974,12916,11025,1,165,8778,172810,
%T A008956 1057221,893025,1,286,28743,1234948,21967231,128816766,108056025,1,455,
%U A008956 77077,6092515,230673443,3841278805,21878089479,18261468225,1,680
%N A008956 Triangle of central factorial numbers 4^k t(2n+1,2n+1-2k).
%C A008956 T(k,n) is the absolute value of the (2n)-th coefficient in Prod[i=1..2k, 
               x+2i-2k-1 ].
%C A008956 Descending row polynomials in x^2 evaluated at k generate odd coefficients 
               of e.g.f. sin(arcsin(kt)/k): 1, x^2 - 1, 9x^4 - 10x^2 + 1, 225x^6 
               - 259x^4 + 34x^2 - 1, ... - Ralf Stephan, Jan 16 2005
%C A008956 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 
               2009: (Start)
%C A008956 We define (Pi/2)*Beta(n-1/2-z/2,n-1/2+z/2)/Beta(n-1/2,n-1/2) = (Pi/2)*Gamma(n-1/
               2-z/2)* Gamma(n-1/2+z/2)/Gamma(n-1/2)^2 = sum(BG2[2m,n]*z^(2m), m 
               = 0..infinity) with Beta(z,w) the Beta function. Our definition leads 
               to BG2[2m,1] = 2*beta(2m+1) and the recurrence relation BG2[2m,n] 
               = BG2[2m,n-1] - BG2[2m-2,n-1]/(2*n-3)^2 for m = -2, -1, 0, 1, 2, 
               .. and n = 2, 3, .. , with beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). 
               We observe that beta(2m+1) = 0 for m = -1, -2, -3, .. .We found for 
               the BG2[2*m,n] = sum((-1)^(k+n)*t2(n-1,k-1)* 2*beta(2*m-2*n+2*k+1),
               k=1..n)/((2*n-3)!!)^2 with the central factorial numbers t2(n,m) 
               as defined above; see also the Maple program.
%C A008956 From the BG2 matrix and the closely related EG2 and ZG2 matrices, see 
               A008955, we arrive at the LG2 matrix which is defined by LG2[2m-1,
               1] = 2*lambda(2*m) and the recurrence relation LG2[2*m-1,n] = LG2[2*m-3,
               n-1]/((2*n-3)*(2*n-1)) - (2*n-3)*LG2[2*m-1,n-1]/(2*n-1) for m = -2, 
               -1, 0, 1, 2, .. and n = 2, 3, .. , with lambda(m) = (1-2^(-m))*zeta(m) 
               with zeta(m) the Riemann zeta function. We found for the matrix coefficients 
               LG2[2m-1,n] = sum((-1)^(k+1)* t2(n-1,k-1)*2*lambda(2*m-2*n+2*k)/((2*n-1)!!*(2*n-3)!!), 
               k=1..n) and we see that the central factorial numbers t2(n,m) once 
               again play a crucial role.
%C A008956 (End)
%D A008956 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A008956 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 A008956 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 A008956 Conjecture row sums: sum_{k=0..n} T(n,k) = |A101927(n+1)|. [From R. J. 
               Mathar (mathar(AT)strw.leidenuniv.nl), May 29 2009]
%F A008956 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 
               2009: (Start)
%F A008956 t2(n,k) = (2*n-1)^2*t2(n-1,k-1)+t2(n-1,k) with t2(n,0) = 1 and t2(n,n)=((2*n-1)!!)^2.
%F A008956 (End)
%e A008956 1; 1,1; 1,10,9; 1,35,259,225; ...
%p A008956 A008956 := proc(n,k) local i ; mul( x+2*i-2*n-1,i=1..2*n) ; expand(%) 
               ; coeftayl(%,x=0,2*(n-k)) ; abs(%) ; end: for n from 0 to 10 do for 
               k from 0 to n do printf("%a,",A008956(n,k)) ; od: od: [From R. J. 
               Mathar (mathar(AT)strw.leidenuniv.nl), May 29 2009]
%p A008956 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 
               2009: (Start)
%p A008956 restart; nmax:=8; for n from 0 to nmax do t2(n,0):=1 od: for n from 0 
               to nmax do t2(n,n):=(doublefactorial(2*n-1))^2 od: for n from 1 to 
               nmax do for m from 1 to n-1 do t2(n,m) := (2*n-1)^2*t2(n-1,m-1)+t2(n-1,
               m) od: od: T:=0: for n from 0 to nmax do for m from 0 to n do a(T):=t2(n,
               m): T:=T+1 od: od: seq(a(n), n=0..T-1);
%p A008956 (End)
%o A008956 (PARI) T(n,k)=if(n<=0,k==0,(-1)^k*polcoeff(numerator(2^(2*n-1)/sum(j=0,
               2*n-1,binomial(2*n-1,j)/(x+2*n-1-2*j))),2*n-2*k))
%Y A008956 Cf. A008958.
%Y A008956 Columns include A000447, A001823. Right-hand columns include A001818, 
               A001824, A001825. Cf. A008955.
%Y A008956 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 18 
               2009: (Start)
%Y A008956 Appears in A160480 (Beta triangle), A160487 (Lambda triangle), A160479 
               (ZL(n) sequence), A161736, A002197 and A002198.
%Y A008956 (End)
%Y A008956 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 
               2009: (Start)
%Y A008956 Cf. A162443 (BG1 matrix) and A162448 (LG1 matrix).
%Y A008956 (End)
%Y A008956 Sequence in context: A147974 A038310 A118768 this_sequence A081101 A022966 
               A023452
%Y A008956 Adjacent sequences: A008953 A008954 A008955 this_sequence A008957 A008958 
               A008959
%K A008956 tabl,nonn,easy
%O A008956 0,5
%A A008956 N. J. A. Sloane (njas(AT)research.att.com).
%E A008956 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 16 2000

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