0,5

T(k,n) is the absolute value of the (2n)-th coefficient in Prod[i=1..2k, x+2i-2k-1 ].

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

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

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.

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.

(End)

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]

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]

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]

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

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.

(End)

1; 1,1; 1,10,9; 1,35,259,225; ...

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]

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

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);

(End)

(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))

Cf. A008958.

Columns include A000447, A001823. Right-hand columns include A001818, A001824, A001825. Cf. A008955.

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

Appears in A160480 (Beta triangle), A160487 (Lambda triangle), A160479 (ZL(n) sequence), A161736, A002197 and A002198.

(End)

Sequence in context: A147974 A038310 A118768 this_sequence A081101 A022966 A023452

tabl,nonn,easy,new

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

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