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A008955 Triangle of central factorial numbers |t(2n,2n-2k)| read by rows. 32
1, 1, 1, 1, 5, 4, 1, 14, 49, 36, 1, 30, 273, 820, 576, 1, 55, 1023, 7645, 21076, 14400, 1, 91, 3003, 44473, 296296, 773136, 518400, 1, 140, 7462, 191620, 2475473, 15291640, 38402064, 25401600, 1, 204, 16422, 669188, 14739153, 173721912 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Discussion of Central Factorial Numbers by N. J. A. Sloane, Feb 01 2011: (Start)

Here is Riordan's definition of the central factorial numbers t(n,k) given in Combinatorial Identities, Section 6.5:

For n >= 0, expand the polynomial

x^[n] = x*Product{i=1..n-1} (x+n/2-i) = Sum_{k=0..n} t(n,k)*x^k.

The t(n,k) are not always integers. The cases n even and n odd are best handled separately.

For n=2m, we have:

x^[2m] = Product_{i=0..m-1} (x^2-i^2) = Sum_{k=1..m} t(2m,2k)*x^(2k).

E.g. x^[8] = x^2(x^2-1^2)(x^2-2^2)(x^2-3^2) = x^8-14x^6+49x^4-36x^2,

which corresponds to row 4 of the present triangle.

So the m-th row of the present triangle gives the absolute values of the coefficients in the expansion of Product_{i=0..m-1} (x^2-i^2).

Equivalently, and simpler, the n-th row gives the coefficients in the expansion of Product_{i=1..n-1}(x+i^2), highest powers first.

For n odd, n=2m+1, we have:

x^[2m+1] = x*Product_{i=0..m-1}(x^2-((2i+1)/2)^2) = Sum_{k=0..m} t(2m+1,2k+1)*x^(2k+1).

E.g. x^[5] = x(x^2-(1/2)^2)(x^2-(3/2)^2) = x^5-10x^3/4+9x/16,

which corresponds to row 2 of the triangle in A008956.

We now rescale to get integers by replacing x by x/2 and multiplying by 2^(2m+1) (getting 1, -10, 9 from the example).

The result is that row m of triangle A008956 gives the coefficients in the expansion of x*Product_{i=0..m} (x^2-(2i+1)^2).

Equivalently, and simpler, the n-th row of A008956 gives the coefficients in the expansion of Product_{i=0..n-1} (x+(2i+1)^2), highest powers first.

Note that the n-th row of A182867 gives the coefficients in the expansion of Product_{i=1..n} (x+(2i)^2), highest powers first.

(End)

Contribution from Johannes W. Meijer, Jun 18 2009: (Start)

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.

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.

(End)

REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part 1, Springer-Verlag 1985.

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.

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

LINKS

T. D. Noe, Rows n=0..50 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.

R. H. Boels, Three particle superstring amplitudes with massive legs, arXiv preprint arXiv:1201.2655 [hep-th], 2012.

R. H. Boels, T. Hansen, String theory in target space, arXiv preprint arXiv:1402.6356 [hep-th], 2014.

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).

M. W. Coffey, M. C. Lettington, On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat, arXiv:1510.05402 [math.NT], 2015.

T. L. Curtright, T. S. Van Kortryk, On Rotations as Spin Matrix Polynomials, arxiv:1408.0767 [math-ph], 2014.

J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.

Mircea Merca, A Special Case of the Generalized Girard-Waring Formula J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

FORMULA

The n-th row gives the coefficients in the expansion of Product_{i=1..n-1}(x+i^2), highest powers first (see Comments section).

The triangle can be obtained fron the recurrence t1(n,k) = n^2*t1(n-1,k-1) + t1(n-1,k) with t1(n,0) = 1 and t1(n,n) = (n!)^2.

t1(n,k) = sum_{j=-k..k} (-1)^j*s(n+1,n+1-k+j)*s(n+1,n+1-k-j) = sum_{j=0..2*(n+1-k)} (-1)^(n+1-k+j)*s(n+1,j)*s(n+1,2*(n+1-k)-j), where s(n,k) are Stirling numbers of the first kind, A048994. - Mircea Merca, Apr 02 2012

E.g.f. cosh(2/sqrt(t)*asin(sqrt(t)*z/2)) = 1 + z^2/2! + (1 + t)*z^4/4! + (1 + 5*t + 4*t^2)*z^6/6! + ... (see Berndt, p.263 and p.306). - Peter Bala, Aug 29 2012

T(n,m) = (2*(n+1))!*sum(k=0..m, (-1)^k*binomial(n,m-k)*sum(i=0..2*k (2^(i-2*m)*stirling1(2*(n-m+1)+i,2*(n-m+1))*binomial(2*(n-m+1)+2*k-1,2*(n-m+1)+i-1))/(2*(n-m+1)+i)!)). - Vladimir Kruchinin, Oct 05 2013

EXAMPLE

1;

1,   1;

1,   5,   4;

1,  14,  49,  36;

1,  30, 273, 820, 576; ...

MAPLE

nmax:=7: for n from 0 to nmax do t1(n, 0):=1: t1(n, n):=(n!)^2 end do: for n from 1 to nmax do for k from 1 to n-1 do t1(n, k) := t1(n-1, k-1)*n^2 + t1(n-1, k) end do: end do: seq(seq(t1(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jun 18 2009, Revised Sep 16 2012

t1 := proc(n, k)

        sum((-1)^j*stirling1(n+1, n+1-k+j)*stirling1(n+1, n+1-k-j), j=-k..k) ;

end proc: # Mircea Merca, Apr 02 2012

MATHEMATICA

t[n_, 0] = 1; t[n_, n_] = (n!)^2; t[n_ , k_ ]:=t[n, k] = n^2 t[n-1, k-1] + t[n-1, k];

Flatten[Table[t[n, k], {n, 0, 8}, {k, 0, n}] ][[1 ;; 42]]

(* Jean-Fran├žois Alcover, May 30 2011, after recurrence formula *)

PROG

(Sage) This triangle is (0, 0)-based.

def A008955(n, k) :

    if k==0 : return 1

    if k==n : return factorial(n)^2

    return n^2*A008955(n-1, k-1) + A008955(n-1, k)

for n in (0..7) : print [A008955(n, k) for k in (0..n)] # Peter Luschny, Feb 04 2012

(Maxima)

T(n, m):=(2*(n+1))!*sum((-1)^k*binomial(n, m-k)*sum((2^(i-2*m)*stirling1(2*(n-m+1)+i, 2*(n-m+1))*binomial(2*(n-m+1)+2*k-1, 2*(n-m+1)+i-1))/(2*(n-m+1)+i)!, i, 0, 2*k), k, 0, m); /* Vladimir Kruchinin, Oct 05 2013 */

(Haskell)

a008955 n k = a008955_tabl !! n !! k

a008955_row n = a008955_tabl !! n

a008955_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 + 1

-- Reinhard Zumkeller, Dec 24 2013

CROSSREFS

Cf. A036969.

Columns include A000330, A000596, A000597. Right-hand columns include A001044, A001819, A001820, A001821. Row sums are in A101686.

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. - Johannes W. Meijer, Jun 18 2009, Jul 06 2009 and Aug 17 2009

Cf. A234324 (central terms).

Sequence in context: A213055 A005752 A098494 * A182824 A152862 A108440

Adjacent sequences:  A008952 A008953 A008954 * A008956 A008957 A008958

KEYWORD

tabl,nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

There's an error in the last column of Riordan's table (change 46076 to 21076).

More terms from Vladeta Jovovic, Apr 16 2000

Link added and cross-references edited by Johannes W. Meijer, Aug 17 2009

Discussion of Riordan's definition of central factorial numbers added by N. J. A. Sloane, Feb 01 2011

STATUS

approved

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Last modified July 21 04:27 EDT 2017. Contains 289632 sequences.