A113025 Triangle of integer coefficients of polynomials P(n,x) of degree n, and falling powers of x, arising in diagonal Padé approximation of exp(x).

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

Offset: 0

Author

Benoit Cloitre, Jan 03 2006

Keywords

• nonn
• tabl
• easy

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)

Examples

			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)
{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)
		

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
• E. Grosswald, Bessel Polynomials: Recurrence Relations, Lecture Notes Math. vol. 698, 1978, p. 18.
• H. L. Krall and Orrin Frink, A New Class of Orthogonal Polynomials: The Bessel Polynomials, Trans. Amer. Math. Soc. 65, 100-115, 1949.
• Eric Weisstein's World of Mathematics, Padé approximants.
• F. Wielonsky, Asymptotics of diagonal Hermite-Pade approximants to exp(x), J. Approx. Theory 90 (1997) 283-298.

Crossrefs

Cf. A001498, A001517, A303986 (signed version).

Programs

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 *)
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 *)

PARI

T(n,k)=(n+k)!/k!/(n-k)!

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