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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).

%I A113025 #36 Feb 16 2025 08:32:59
%S A113025 1,1,2,1,6,12,1,12,60,120,1,20,180,840,1680,1,30,420,3360,15120,30240,
%T A113025 1,42,840,10080,75600,332640,665280,1,56,1512,25200,277200,1995840,
%U A113025 8648640,17297280,1,72,2520,55440,831600,8648640,60540480,259459200
%N 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).
%C A113025 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)).
%C A113025 From _Roger L. Bagula_, Feb 15 2009: (Start)
%C A113025 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]
%C A113025 P(n, x) = Sum_{k=0..n} (n+k)!/(k!*(n-k)!)*x^(n-k).
%C A113025 Row sums are A001517. (End)
%D A113025 J. Riordan, Combinatorial Identities, Wiley, 1968, p.77, 10. [From _Roger L. Bagula_, Feb 15 2009]
%H A113025 G. C. Greubel, <a href="/A113025/b113025.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A113025 E. Grosswald, <a href="http://dx.doi.org/10.1007/BFb0063138">Bessel Polynomials: Recurrence Relations</a>, Lecture Notes Math. vol. 698, 1978, p. 18.
%H A113025 H. L. Krall and Orrin Frink, <a href="https://doi.org/10.1090/S0002-9947-1949-0028473-1">A New Class of Orthogonal Polynomials: The Bessel Polynomials</a>, Trans. Amer. Math. Soc. 65, 100-115, 1949.
%H A113025 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PadeApproximant.html">Padé approximants</a>.
%H A113025 F. Wielonsky, <a href="https://doi.org/10.1006/jath.1996.3081">Asymptotics of diagonal Hermite-Pade approximants to exp(x)</a>, J. Approx. Theory 90 (1997) 283-298.
%F A113025 From _Wolfdieter Lang_, May 11 2018: (Start)
%F A113025 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).
%F A113025 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).
%F A113025 (End)
%F A113025 T(n, k) = Pochhammer(n+1, k)*binomial(n, k). # _Peter Luschny_, May 11 2018
%e A113025 P(3,x) = x^3 + 12*x^2 + 60*x + 120.
%e A113025 y_3(2*x) = 1 + 12*x + 60*x^2 + 120*x^3. (Bessel with x -> 2*x).
%e A113025 From _Roger L. Bagula_, Feb 15 2009: (Start)
%e A113025 {1},
%e A113025 {1, 2},
%e A113025 {1, 6, 12},
%e A113025 {1, 12, 60, 120},
%e A113025 {1, 20, 180, 840, 1680},
%e A113025 {1, 30, 420, 3360, 15120, 30240},
%e A113025 {1, 42, 840, 10080, 75600, 332640, 665280},
%e A113025 {1, 56, 1512, 25200, 277200, 1995840, 8648640, 17297280},
%e A113025 {1, 72, 2520, 55440, 831600, 8648640, 60540480, 259459200, 518918400},
%e A113025 {1, 90, 3960, 110880, 2162160, 30270240, 302702400, 2075673600, 8821612800, 17643225600},
%e A113025 {1, 110, 5940, 205920, 5045040, 90810720, 1210809600, 11762150400, 79394515200, 335221286400, 670442572800} (End)
%p A113025 T := (n, k) -> pochhammer(n+1, k)*binomial(n, k):
%p A113025 seq(seq(T(n, k), k=0..n), n=0..9); # _Peter Luschny_, May 11 2018
%t A113025 L[n_, m_] = (n + m)!/((n - m)!*m!);
%t A113025 Table[Table[L[n, m], {m, 0, n}], {n, 0, 10}];
%t A113025 Flatten[%] (* _Roger L. Bagula_, Feb 15 2009 *)
%t A113025 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 *)
%o A113025 (PARI) T(n,k)=(n+k)!/k!/(n-k)!
%Y A113025 Cf. A001498, A001517, A303986 (signed version).
%K A113025 nonn,tabl,easy
%O A113025 0,3
%A A113025 _Benoit Cloitre_, Jan 03 2006