A370707 - OEIS

%I #40 Mar 07 2024 10:46:22

%S 1,1,1,1,4,12,1,9,72,360,1,16,240,2880,20160,1,25,600,12600,201600,

%T 1814400,1,36,1260,40320,1088640,21772800,239500800,1,49,2352,105840,

%U 4233600,139708800,3353011200,43589145600,1,64,4032,241920,13305600,638668800,24908083200,697426329600,10461394944000

%N Triangle read by rows: T(n, k) = (-1)^k*Product_{j=0..k-1} (j - n)*(j + n), for 0 <= k <= n.

%C The definition, and also the representation T(n, k) = ff(n, k) * rf(n, k) (see the first formula), makes it natural to call this triangle the central factorial numbers.

%F T(n, k) = FallingFactorial(n, k) * RisingFactorial(n, k).

%F T(n, k) = (n*(n + k - 1)!)/(n - k)! if k > 0, and T(n, 0) = 1.

%F Calling the numbers in the second formula cf leads to the memorable form cf(n, k) = ff(n, k) * rf(n, k). This identity generalizes to the function

%F   cf(x, n) = x*Gamma(x + n)/Gamma(x - n + 1) for n > 0 and cf(x, 0) = 1.

%F The last equation shows that the variable 'n' does not have to be an integer but can be any complex number if only the quotient remains defined (which one often can achieve by taking the limit). Indeed, in the classical Steffensen-Riordan case, n/2 is used instead of n, which leads to the complex situation Sloane discusses in A008955.

%F T(n, k) = -n*Pochhammer(1 - n - k, 2*k - 1) for n > 0.

%F T(n, k) = k!*binomial(n, k)*Pochhammer(n, k) = k!*A370706(n, k).

%F T(n, n) = n!*Pochhammer(n, n) (valid for n >= 0, whereas T(n, n) = (2*n)!/2 = A002674(n) is valid for n >= 1 only).

%F T(n, k) = T(n, k - 1)*(n^2 - (k - 1)^2) if k > 0, otherwise 1. (Recurrence)

%F The cf(n, k) are values of the polynomials Pcf(n, x) = Product_{k=0..n-1} (x^2 - k^2), whose coefficients vanish for odd powers and for even powers are A269944.

%F T(n, k) = Pcf(k, n) where Pcf(k,x) = Sum_{j=0..k) (-1)^(k-j)*A269944(k,j)*x^(2*j).

%F The central factorials can be described in three different ways: By the product T(n, k) = f(n, k) * rf(n, k), by the complex function cf(x, n), and through the polynomials Pcf(n, x). Although these relations are self-contained, they are regarded as only one-half of a more general notion, namely as central factorials of the first kind.

%F There is a fundamental connection with the Stirling numbers of first kind (A048994). The easiest way to see this is to generalize the definition: Let CF(z, s) = Product_{j=0..n-1} (z - s(j)), where s(j) is some complex sequence. Then the coefficients of CF(z, s) are equal to the Stirling_1 numbers if s = 0, 1, 2, ..., n, ..., and they are equal to the coefficients of our Pcf(n, z) polynomials if s = 0, 1, 4, ..., n^2, .... (This is also why A269944 is called the 'Stirling cycle numbers of order 2'. For completeness, if s = 1, 1, 1, ..., then the coefficients of CF(z, s), the 'Stirling cycle numbers of order 0', are the signed Pascal triangle A130595. See A269947 for order 3.)

%e Triangle starts:

%e   [0] 1;

%e   [1] 1,  1;

%e   [2] 1,  4,   12;

%e   [3] 1,  9,   72,    360;

%e   [4] 1, 16,  240,   2880,   20160;

%e   [5] 1, 25,  600,  12600,  201600,   1814400;

%e   [6] 1, 36, 1260,  40320, 1088640,  21772800,  239500800;

%e   [7] 1, 49, 2352, 105840, 4233600, 139708800, 3353011200, 43589145600;

%e .

%e T(n, k) is a product where 'n' is the 'center' and 'k' is the 'half-length' of the product. For instance, T(5, 4) = (5-3)*(5-2)*(5-1)*5 * 5*(5+1)*(5+2)*(5+3) = 201600. Now consider the polynomial P(4, x) = -36*x^2 + 49*x^4 - 14*x^6 + x^8. Evaluating this polynomial at x = 5 shows P(4, 5) = 201600 = T(5, 4). The coefficients of the polynomial are row 4 of A269944.

%p T := (n, k) -> local j; (-1)^k * mul((j - n)*(j + n), j = 0..k-1):

%p seq(seq(T(n, k), k = 0..n), n = 0..8);

%p # The central factorial numbers:

%p cf := (n, k) -> ifelse(k = 0, 1, n*(n + k - 1)! / (n - k)! ):

%p for n from 0 to 6 do seq(cf(n, k), k = 0..n) od;

%p # Alternative (recurrence):

%p T := proc(n, k) option remember;

%p if k = 0 then 1 else T(n, k - 1)*(n^2 - (k - 1)^2) fi end:

%p for n from 0 to 7 do seq(T(n, k), k = 0..n) od;

%p # Illustrating the connection with the cf-polynomials and their coefficients:

%p cfpoly := (n,x) -> local k; mul(x^2 - k^2, k = 0..n-1):

%p A370707row := n -> local k; [seq(cfpoly(k, n), k = 0..n)]:

%p A204579row := n -> local k; [seq(coeff(cfpoly(n, x), x, 2*k), k = 0..n)]:

%p for n from 0 to 5 do lprint([n], A370707row(n), A204579row(n)) od;

%t T[n_, k_] := If[n == 0, 1, -n Pochhammer[1 - n - k, 2 k - 1]];

%t Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

%o (SageMath)

%o def T(n, k): return falling_factorial(n, k) * rising_factorial(n, k)

%o for n in range(9): print([T(n, k) for k in range(n + 1)])

%o (Python)

%o from math import prod

%o def T(n, k): return (-1)**k * prod((j - n)*(j + n) for j in range(k))

%o print([T(n, k) for n in range(8) for k in range(n + 1)])

%Y Diagonals: A002674, A327882.

%Y Columns: A000290, A047928.

%Y Cf. A370704 (row sums), A370706, A094728, A048994 (Stirling1), A130595 (order 0), A269947 (order 3)

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Feb 27 2024