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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). 6
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 (list; table; graph; refs; listen; history; text; internal format)
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)

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 Fink, 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.

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)

{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 *)

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

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

Sequence in context: A283746 A049949 A106192 * A113216 A303986 A081064

Adjacent sequences:  A113022 A113023 A113024 * A113026 A113027 A113028

KEYWORD

nonn,tabl,easy

AUTHOR

Benoit Cloitre, Jan 03 2006

STATUS

approved

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Last modified May 28 01:09 EDT 2018. Contains 304726 sequences. (Running on oeis4.)