OFFSET
0,3
COMMENTS
exp(x) is well approximated by P(n,x)/P(n,-x). (P(n,1)/P(n,-1))_{n>=0} is a sequence of convergents to e: i.e., P(n,1) = A001517(n) and P(n,-1) = abs(A002119(n)).
From Roger L. Bagula, Feb 15 2009: (Start)
The row polynomials in rising powers of x are y_n(2*x) = Sum_{k=0..n} binomial(n+k, 2*k)*((2*k)!/k!)*x^k, for n >= 0, with the Bessel polynomials y_n(x) of Krall and Frink, eq. (3), (see also Grosswald, p. 18, eq. (7) and Riordan, p. 77). For the coefficients see A001498. [Edited by Wolfdieter Lang, May 11 2018]
P(n, x) = Sum_{k=0..n} (n+k)!/(k!*(n-k)!)*x^(n-k).
Row sums are A001517. (End)
From Peter Bala, Feb 12 2026: (Start)
Let Q(n,x) be the polynomial obtained by changing the sign of the coefficients of the terms x^(4*n+2) and x^(4*n+3) in the polynomial P(n,x), that is, Q(n,x) = Re(P(n,i*x)) + Im(P(n,i*x)), where i = sqrt(-1). Then Q(n,x)/Q(n,-x) is the diagonal (n/n) Padé approximant to tan(x) + sec(x) (follows from tan(x) + sec(x) = (1 + tan(x/2))/(1 - tan(x/2)) and tan(x/2) = (1/i)*(exp(i*x) - 1)/(exp(i*x) + 1)).
For example, the (5/5) Padé approximant to tan(x) + sec(x) is (x^5 + 30*x^4 - 420*x^3 - 3360*x^2 + 15120*x + 30240)/(-x^5 + 30*x^4 + 420*x^3 - 3360*x^2 - 15120*x + 30240). (End)
REFERENCES
J. Riordan, Combinatorial Identities, Wiley, 1968, p.77, 10. [From Roger L. Bagula, Feb 15 2009]
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Emil Grosswald, Bessel Polynomials: Recurrence Relations, Lecture Notes Math. 698 (1978), p. 18.
Bulat Khaidarov, Enumeration of Certain k-Block Partitions of Barry, J. Int. Seq. 29 (2026), Art. 26.2.8. See p. 6 (Table 2).
H. L. Krall and Orrin Frink, A New Class of Orthogonal Polynomials: The Bessel Polynomials, Trans. Amer. Math. Soc. 65 (1949), 100-115.
Eric Weisstein's World of Mathematics, Padé approximants.
Franck Wielonsky, Asymptotics of diagonal Hermite-Pade approximants to exp(x), J. Approx. Theory 90 (1997), 283-298.
FORMULA
From Wolfdieter Lang, May 11 2018: (Start)
T(n, k) = binomial(n+k, 2*k)*(2*k)!/k! = (n+k)!/((n-k)!*k!), n >= 0, k = 0..n. (see the R. L. Baluga comment above).
Recurrence (adapted from A001498, see the Grosswald reference): For n >= 0, k = 0..n: a(n, k) = 0 for n < k (zeros not shown in the triangle), a(n, -1) = 0, a(0, 0) = 1 = a(1, 0) and otherwise a(n, k) = 2*(2*n-1)*a(n-1, k-1) + a(n-2, k).
(End)
T(n, k) = Pochhammer(n+1, k)*binomial(n, k). # Peter Luschny, May 11 2018
EXAMPLE
P(3,x) = x^3 + 12*x^2 + 60*x + 120.
y_3(2*x) = 1 + 12*x + 60*x^2 + 120*x^3. (Bessel with x -> 2*x).
From Roger L. Bagula, Feb 15 2009: (Start)
Triangle begins:
1;
1, 2;
1, 6, 12;
1, 12, 60, 120;
1, 20, 180, 840, 1680;
1, 30, 420, 3360, 15120, 30240;
1, 42, 840, 10080, 75600, 332640, 665280;
1, 56, 1512, 25200, 277200, 1995840, 8648640, 17297280;
1, 72, 2520, 55440, 831600, 8648640, 60540480, 259459200, 518918400;
1, 90, 3960, 110880, 2162160, 30270240, 302702400, 2075673600, 8821612800, 17643225600;
1, 110, 5940, 205920, 5045040, 90810720, 1210809600, 11762150400, 79394515200, 335221286400, 670442572800;
... (End)
MAPLE
T := (n, k) -> pochhammer(n+1, k)*binomial(n, k):
seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, May 11 2018
MATHEMATICA
L[n_, m_] = (n + m)!/((n - m)!*m!);
Table[Table[L[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%] (* Roger L. Bagula, Feb 15 2009 *)
(* Alternative: *)
P[x_, n_] := Sum[ (2*n - k)!/(k!*(n - k)!)*x^(k), {k, 0, n}]; Table[Reverse[CoefficientList[P[x, n], x]], {n, 0, 10}] // Flatten (* G. C. Greubel, Aug 15 2017 *)
PROG
(PARI) T(n, k)=(n+k)!/k!/(n-k)!
CROSSREFS
KEYWORD
AUTHOR
Benoit Cloitre, Jan 03 2006
STATUS
approved