A008956 Triangle of central factorial numbers |4^k t(2n+1,2n+1-2k)| read by rows (n>=0, k=0..n).

1, 1, 1, 1, 10, 9, 1, 35, 259, 225, 1, 84, 1974, 12916, 11025, 1, 165, 8778, 172810, 1057221, 893025, 1, 286, 28743, 1234948, 21967231, 128816766, 108056025, 1, 455, 77077, 6092515, 230673443, 3841278805, 21878089479, 18261468225, 1, 680

Offset: 0

Author

N. J. A. Sloane

Keywords

• tabl
• nonn
• easy

Comments

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

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

From Johannes W. Meijer, 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)

Examples

			Triangle begins:
[1]
[1, 1]
[1, 10, 9]
[1, 35, 259, 225]
[1, 84, 1974, 12916, 11025]
[1, 165, 8778, 172810, 1057221, 893025]
[1, 286, 28743, 1234948, 21967231, 128816766, 108056025]
[1, 455, 77077, 6092515, 230673443, 3841278805, 21878089479, 18261468225]
...
		

References

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

Links

• Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
• 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_, Jun 18 2009]
• R. H. Boels, T. Hansen, String theory in target space, arXiv preprint arXiv:1402.6356 [hep-th], 2014.
• T. L. Curtright, D. B. Fairlie, C. K. Zachos, A compact formula for rotations as spin matrix polynomials, arXiv preprint arXiv:1402.3541 [math-ph], 2014.
• T. L. Curtright, T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arXiv:1408.0767 [math-ph], 2014.
• M. Eastwood and H. Goldschmidt, Zero-energy fields on complex projective space, arXiv preprint arXiv:1108.1602 [math.DG], 2011.
• M. Eastwood, The X-ray transform on projective space. - From _N. J. A. Sloane_, Oct 22 2012

Crossrefs

Cf. A008958.

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

Appears in A160480 (Beta triangle), A160487 (Lambda triangle), A160479 (ZL(n) sequence), A161736, A002197 and A002198. - Johannes W. Meijer, Jun 18 2009

Cf. A162443 (BG1 matrix) and A162448 (LG1 matrix). - Johannes W. Meijer, Jul 06 2009

Cf. A001147.

Programs

Haskell

a008956 n k = a008956_tabl !! n !! k
a008956_row n = a008956_tabl !! n
a008956_tabl = [1] : f [1] 1 1 where
   f xs u t = ys : f ys v (t * v) where
     ys = zipWith (+) (xs ++ [t^2]) ([0] ++ map (* u^2) (init xs) ++ [0])
     v = u + 2
-- Reinhard Zumkeller, Dec 24 2013

Maple

f:=n->mul(x+(2*i+1)^2,i=0..n-1);
for n from 0 to 12 do
t1:=eval(f(n)); t1d:=degree(t1);
t12:=y^t1d*subs(x=1/y,t1); t2:=seriestolist(series(t12,y,20));
lprint(t2);
od: # N. J. A. Sloane, Feb 01 2011
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
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

Mathematica

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 *)

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 */

Formula

Conjecture row sums: Sum_{k=0..n} T(n,k) = |A101927(n+1)|. - R. J. Mathar, May 29 2009

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

Extensions

More terms from Vladeta Jovovic, Apr 16 2000

Edited by N. J. A. Sloane, Feb 01 2011