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%I #63 Aug 24 2020 22:40:02
%S 1,1,1,1,10,9,1,35,259,225,1,84,1974,12916,11025,1,165,8778,172810,
%T 1057221,893025,1,286,28743,1234948,21967231,128816766,108056025,1,
%U 455,77077,6092515,230673443,3841278805,21878089479,18261468225,1,680
%N Triangle of central factorial numbers |4^k t(2n+1,2n+1-2k)| read by rows (n>=0, k=0..n).
%C The n-th row gives the coefficients in the expansion of Product_{i=0..n-1} (x+(2i+1)^2), highest powers first (see the discussion of central factorial numbers in A008955). - _N. J. A. Sloane_, Feb 01 2011
%C 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 From _Johannes W. Meijer_, Jun 18 2009: (Start)
%C 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 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 (End)
%D P. L. Butzer, M. Schmidt, E. L. Stark and 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_, Jun 18 2009]
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%H Reinhard Zumkeller, <a href="/A008956/b008956.txt">Rows n = 0..100 of triangle, flattened</a>
%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. [From _Johannes W. Meijer_, Jun 18 2009]
%H R. H. Boels, T. Hansen, <a href="http://arxiv.org/abs/1402.6356">String theory in target space</a>, arXiv preprint arXiv:1402.6356 [hep-th], 2014.
%H T. L. Curtright, D. B. Fairlie, C. K. Zachos, <a href="http://arxiv.org/abs/1402.3541">A compact formula for rotations as spin matrix polynomials</a>, arXiv preprint arXiv:1402.3541 [math-ph], 2014.
%H T. L. Curtright, T. S. Van Kortryk, <a href="http://arxiv.org/abs/1408.0767">On Rotations as Spin Matrix Polynomials</a>, arXiv:1408.0767 [math-ph], 2014.
%H M. Eastwood and H. Goldschmidt, <a href="http://arxiv.org/abs/1108.1602">Zero-energy fields on complex projective space</a>, arXiv preprint arXiv:1108.1602 [math.DG], 2011.
%H M. Eastwood, <a href="https://maths-people.anu.edu.au/~eastwood/fayetteville5.pdf">The X-ray transform on projective space</a>. - From _N. J. A. Sloane_, Oct 22 2012
%F Conjecture row sums: Sum_{k=0..n} T(n,k) = |A101927(n+1)|. - _R. J. Mathar_, May 29 2009
%F May be generated by the recurrence 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. - _Johannes W. Meijer_, Jun 18 2009
%e Triangle begins:
%e [1]
%e [1, 1]
%e [1, 10, 9]
%e [1, 35, 259, 225]
%e [1, 84, 1974, 12916, 11025]
%e [1, 165, 8778, 172810, 1057221, 893025]
%e [1, 286, 28743, 1234948, 21967231, 128816766, 108056025]
%e [1, 455, 77077, 6092515, 230673443, 3841278805, 21878089479, 18261468225]
%e ...
%p f:=n->mul(x+(2*i+1)^2,i=0..n-1);
%p for n from 0 to 12 do
%p t1:=eval(f(n)); t1d:=degree(t1);
%p t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
%p lprint(t2);
%p od: # _N. J. A. Sloane_, Feb 01 2011
%p 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: # _R. J. Mathar_, May 29 2009
%p nmax:=7: for n from 0 to nmax do t2(n, 0):=1: t2(n, n):=(doublefactorial(2*n-1))^2 od: for n from 1 to nmax do for k from 1 to n-1 do t2(n, k) := (2*n-1)^2*t2(n-1, k-1)+t2(n-1, k) od: od: seq(seq(t2(n, k), k=0..n), n=0..nmax); # _Johannes W. Meijer_, Jun 18 2009, Revised Sep 16 2012
%t t[_, 0] = 1; t[n_, n_] := t[n, n] = ((2*n-1)!!)^2; t[n_, k_] := t[n, k] = (2*n-1)^2*t[n-1, k-1] + t[n-1, k]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 07 2014, after _Johannes W. Meijer_ *)
%o (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))}; /* _Michael Somos_, Feb 24 2003 */
%o (Haskell)
%o a008956 n k = a008956_tabl !! n !! k
%o a008956_row n = a008956_tabl !! n
%o a008956_tabl = [1] : f [1] 1 1 where
%o f xs u t = ys : f ys v (t * v) where
%o ys = zipWith (+) (xs ++ [t^2]) ([0] ++ map (* u^2) (init xs) ++ [0])
%o v = u + 2
%o -- _Reinhard Zumkeller_, Dec 24 2013
%Y Cf. A008958.
%Y Columns include A000447, A001823. Right-hand columns include A001818, A001824, A001825. Cf. A008955.
%Y Appears in A160480 (Beta triangle), A160487 (Lambda triangle), A160479 (ZL(n) sequence), A161736, A002197 and A002198. - _Johannes W. Meijer_, Jun 18 2009
%Y Cf. A162443 (BG1 matrix) and A162448 (LG1 matrix). - _Johannes W. Meijer_, Jul 06 2009
%Y Cf. A001147.
%K tabl,nonn,easy
%O 0,5
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Apr 16 2000
%E Edited by _N. J. A. Sloane_, Feb 01 2011
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