0,3

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))

Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 15 2009: (Start)

Riordan calls these coefficients of Bessel polynomials and gives an alternate form: p(x,n)=Sum[Binomial[n+k,2*k]*((2*k)!/(k!*2^k))*x^k,{k,0,n}].

Row sums are A001517 (End)

F. Wielonsky, Asymptotics of diagonal Hermite-Pade approximants to exp(x), J. Approx. Theory 90 (1997) 283-298.

J. Riordan, Combinatorial Identities, Wiley, 1968, p.77. [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 15 2009]

Frink, O. and H. L. Krall, A new class of orthogonal polynomials, Trans. Amer. Math. Soc. 65,100-115, 1945 [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 15 2009]

E. Weisstein, Pade approximants.

P(n,x)=sum(k=0, n, (n+k)!/k!/(n-k)!*x^(n-k))

P(3,x)=x^3+12*x^2+60*x+120

Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), 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)

Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 15 2009: (Start)

L[n_, m_] = (n + m)!/((n - m)!*m!);

Table[Table[L[n, m], {m, 0, n}], {n, 0, 10}];

Flatten[%] (End)

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

Sequence in context: A039795 A049949 A106192 this_sequence A113216 A081064 A128534

Adjacent sequences: A113022 A113023 A113024 this_sequence A113026 A113027 A113028

nonn,tabl

Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 03 2006