%I #119 Jul 11 2023 12:04:09

%S 1,1,1,1,5,4,1,14,49,36,1,30,273,820,576,1,55,1023,7645,21076,14400,1,

%T 91,3003,44473,296296,773136,518400,1,140,7462,191620,2475473,

%U 15291640,38402064,25401600,1,204,16422,669188,14739153,173721912,1017067024,2483133696,1625702400

%N Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.

%C Discussion of Central Factorial Numbers by _N. J. A. Sloane_, Feb 01 2011: (Start)

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

%C For n >= 0, expand the polynomial

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

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

%C For n=2m, we have:

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

%C 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,

%C which corresponds to row 4 of the present triangle.

%C 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).

%C 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.

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

%C 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).

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

%C which corresponds to row 2 of the triangle in A008956.

%C 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).

%C 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).

%C 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.

%C 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.

%C (End)

%C Contribution from _Johannes W. Meijer_, Jun 18 2009: (Start)

%C 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.

%C 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_{k=1..n} (-1)^(k+1)*t1(n-1,k-1)* 2* zeta(2*m-2*n+2*k)/((n-1)!*(n)!), and we see that the central factorial numbers t1(n,m) once again play a crucial role.

%C (End)

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

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

%H Alois P. Heinz, <a href="/A008955/b008955.txt">Rows n = 0..100, flattened</a> (first 51 rows from T. D. Noe)

%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.

%H R. H. Boels, <a href="http://arxiv.org/abs/1201.2655">Three particle superstring amplitudes with massive legs</a>, arXiv preprint arXiv:1201.2655 [hep-th], 2012.

%H R. H. Boels and 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 P. L. Butzer, M. Schmidt, E. L. Stark and L. Vogt, <a href="http://dx.doi.org/10.1080/01630568908816313">Central Factorial Numbers: Their main properties and some applications</a>, Numerical Functional Analysis and Optimization, 10 (5&6), 419-488 (1989).

%H M. W. Coffey and M. C. Lettington, <a href="http://arxiv.org/abs/1510.05402">On Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber's Formula and some Theorems of Fermat</a>, arXiv:1510.05402 [math.NT], 2015.

%H T. L. Curtright and 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 P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

%H J. W. Meijer and N. H. G. Baken, <a href="http://dx.doi.org/10.1016/0167-7152(87)90041-1">The Exponential Integral Distribution</a>, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.

%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca2/merca7.html">A Special Case of the Generalized Girard-Waring Formula</a>, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.

%H S. Shadrin, L. Spitz, and D. Zvonkine, <a href="https://doi.org/10.1112/jlms/jds010">On double Hurwitz numbers with completed cycles</a>, J. Lond. Math. Soc., II. Ser. 86, No. 2, 407-432 (2012), Corollary 7.5.

%F 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).

%F The triangle can be obtained from 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.

%F 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

%F 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

%F 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

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 5, 4;

%e 1, 14, 49, 36;

%e 1, 30, 273, 820, 576;

%e ...

%p 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

%p t1 := proc(n,k)

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

%p end proc: # _Mircea Merca_, Apr 02 2012

%p # third Maple program:

%p T:= proc(n, k) option remember; `if`(k=0, 1,

%p add(T(j-1, k-1)*j^2, j=1..n))

%p end:

%p seq(seq(T(n, k), k=0..n), n=0..8); # _Alois P. Heinz_, Feb 19 2022

%t 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]]

%t (* _Jean-François Alcover_, May 30 2011, after recurrence formula *)

%o (Sage) # This triangle is (0,0)-based.

%o def A008955(n, k) :

%o if k==0 : return 1

%o if k==n : return factorial(n)^2

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

%o for n in (0..7) : print([A008955(n, k) for k in (0..n)]) # _Peter Luschny_, Feb 04 2012

%o (Maxima)

%o 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 */

%o (Haskell)

%o a008955 n k = a008955_tabl !! n !! k

%o a008955_row n = a008955_tabl !! n

%o a008955_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 + 1

%o -- _Reinhard Zumkeller_, Dec 24 2013

%o (PARI) T(n,k)=if(k==0,1, if(k==n, (n!)^2, n^2*T(n-1, k-1) + T(n-1, k)));

%o for(n=0,8, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Sep 14 2019

%o (Magma)

%o T:= func< n,k | Factorial(2*(n+1))*(&+[(-1)^j*Binomial(n,k-j)*(&+[2^(m-2*k)*StirlingFirst(2*(n-k+1)+m, 2*(n-k+1))*Binomial(2*(n-k+1)+2*j-1, 2*(n-k+1)+m-1)/Factorial(2*(n-k+1)+m): m in [0..2*j]]): j in [0..k]]) >;

%o [T(n,k): k in [0..n], n in [0..8]]; // _G. C. Greubel_, Sep 14 2019

%o (GAP)

%o T:= function(n,k)

%o if k=0 then return 1;

%o elif k=n then return (Factorial(n))^2;

%o else return n^2*T(n-1,k-1) + T(n-1,k);

%o fi;

%o end;

%o Flat(List([0..8], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Sep 14 2019

%Y Cf. A036969.

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

%Y 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

%Y Cf. A234324 (central terms).

%K tabl,nonn,nice,easy

%O 0,5

%A _N. J. A. Sloane_

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

%E More terms from _Vladeta Jovovic_, Apr 16 2000

%E Link added and cross-references edited by _Johannes W. Meijer_, Aug 17 2009

%E Discussion of Riordan's definition of central factorial numbers added by _N. J. A. Sloane_, Feb 01 2011