A370705 Triangle read by rows: T(n, k) = numerator(CF(n, k)) where CF(n, k) = n! * [x^k] [t^n] (t/2 + sqrt(1 + (t/2)^2))^(2*x).

1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 9, 0, -5, 0, 1, 0, 0, 4, 0, -5, 0, 1, 0, -225, 0, 259, 0, -35, 0, 1, 0, 0, -36, 0, 49, 0, -14, 0, 1, 0, 11025, 0, -3229, 0, 987, 0, -21, 0, 1, 0, 0, 576, 0, -820, 0, 273, 0, -30, 0, 1

Offset: 0

Author

Peter Luschny, Mar 02 2024

Keywords

• sign
• tabl
• frac

Comments

The rational triangle R(n, k) contains the central factorial numbers. The central factorial of the first kind is the even subtriangle of R, while the central factorial of the second kind is the odd subtriangle. Since the terms of the even subtriangle can be seen as integers, the rational nature of these numbers is generally disregarded. The denominators of the central factorial of second kind are powers of 4; therefore, they are often studied as integers in the form 4^(n-k)*R(n, k). We will refer to the subtriangles by CF1(n, k) and CF2(n, k).

We recall that if T(n, k) is a number triangle (0 <= k <= n) then

Teven(n, k) = [T(n, k), k=0..n step 2), n=0..len step 2]

is the even subtriangle of T, and the odd subtriangle of T is

Todd(n, k) = [T(n, k), k=1..n step 2), n=1..len step 2], where

'k=a..o step s' denotes the subrange [a, a+s, a+2*s, ..., a+s*floor((o-a)/s)].

The central factorial numbers have their origins in approximation theory and numerical mathematics. They were undoubtedly used for a long time when J. F. Steffensen used them to construct quadrature formulas and presented them in 1924 at the 7th ICM. Four decades later, Carlitz and Riordan adopted the idea for use in combinatorics. While Steffensen originally referred to the numbers as "central differences of nothing," the second part of the name was later omitted.

Examples

			Triangle starts:
[0] 1;
[1] 0,     1;
[2] 0,     0,   1;
[3] 0,    -1,   0,     1;
[4] 0,     0,  -1,     0,  1;
[5] 0,     9,   0,    -5,  0,   1;
[6] 0,     0,   4,     0, -5,   0,   1;
[7] 0,  -225,   0,   259,  0, -35,   0,   1;
[8] 0,     0, -36,     0, 49,   0, -14,   0, 1;
[9] 0, 11025,   0, -3229,  0, 987,   0, -21, 0, 1;
		

References

• Johan Frederik Steffensen, On a class of quadrature formulae. Proceedings of the International Mathematical Congress Toronto 1924, Vol 2, pp. 837-844.

Links

• P. L. Butzer, M. Schmidt, E. L. Stark and L. Vogt. Central factorial numbers; their main properties and some applications, Num. Funct. Anal. Optim., 10 (1989) 419-488.
• Leonard Carlitz and John Riordan, The Divided Central Differences of Zero, Canadian Journal of Mathematics, Volume 15, 1963, pp. 94-100.
• 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.
• Johan Frederik Steffensen, On the Definition of the Central Factorial, Journal of the Institute of Actuaries, Volume 64, Issue 2, July 1933, pp. 165-168.

Crossrefs

See the discussion by Sloane in A008955 of Riordan's notation. In particular, the notation 'T' below does not refer to the present triangle.

Central factorials (rational, general case): (this triangle) / A370703;

t(2n, 2k) (first kind, 'even case') A204579; (signed, T(n, 0) missing)

|t(2n, 2k)| A269944; (unsigned, T(n, 0) = 0^n)

|t(2n, 2n-2k)| A008955;

|t(2n+1, 2n+1-2k)|*4^k A008956;

T(2n, 2k) (second kind, 'odd case') A269945, A036969;

T(2n+1, 2k+1)*4^(n-k) A160562.

Programs

Maple

gf := (t/2 + sqrt(1 + (t/2)^2))^(2*x): ser := series(gf, t, 20):
ct := n -> n!*coeff(ser, t, n):
T := (n, k) -> numer(coeff(ct(n), x, k)):
seq(seq(T(n, k), k = 0..n), n = 0..10);
# Filtering the central factorials of the first resp. second kind:
CF1 := (T,len) -> local n,k; seq(print(seq(T(n,k), k=0..n, 2)), n = 0..len, 2);
CF2 := (T,len) -> local n,k; seq(print(seq(T(n,k), k=1..n, 2)), n = 1..len, 2);