A137338 Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), ordered by rising powers.
1, -1, 1, 0, -3, 1, 3, 6, -6, 1, -12, -9, 26, -10, 1, 45, 3, -109, 71, -15, 1, -198, 81, 501, -475, 155, -21, 1, 1071, -786, -2663, 3329, -1455, 295, -28, 1, -6984, 6711, 16510, -25495, 13729, -3647, 511, -36, 1, 53217, -60309, -117912, 216004, -135961, 43897, -7994, 826, -45, 1, -462330, 589197, 953711
Offset: 1
Examples
[0] 1, [1] -1, 1, [2] 0, -3, 1, [3] 3, 6, -6, 1, [4] -12, -9, 26, -10, 1, [5] 45, 3, -109, 71, -15, 1, [6] -198, 81, 501, -475, 155, -21, 1, [7] 1071, -786, -2663, 3329, -1455, 295, -28, 1, [8] -6984, 6711, 16510, -25495, 13729, -3647, 511, -36, 1, [9] 53217, -60309, -117912, 216004, -135961, 43897, -7994, 826, -45, 1.
Links
- Carl V. L. Charlier, Über die Darstellung willkürlicher Funktionen, Arkiv För Matematik, Astronomi Och Fysik, Band 2, No. 20 (Meddelande från Lunds Astronomiska Observatorium, Series I, No. 27), 1905, 1-35. [Accessible only in the USA via the HathiTrust Digital Library.]
- M. Dunster, Uniform asymptotic expansions for Charlier polynomials, J. Approx. Theory, 112 (2001), pp. 93-133.
- Chun-Hua Ou, Global Asymptotics of the Charlier Polynomials via the Riemann-Hilbert Method.
- Wikipedia, Carl Charlier.
- Wikipedia, Charlier polynomials.
Crossrefs
Cf. A216916.
Programs
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Maple
with(PolynomialTools): C := (n, x) -> if n>0 then expand((x-n)*C(n-1,x)-n*C(n-2,x)) elif n = 0 then 1 else 0 fi: A137338_row := n -> CoefficientList(C(n,x), x); for n from 0 to 7 do A137338_row(n) od; # Peter Luschny, Sep 21 2012
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Mathematica
Ca[x, -1] = 0; Ca[x, 0] = 1; Ca[x_, n_] := Ca[x, n] = (x - (n - 1) - 1)*Ca[x, n - 1] - n*Ca[x, n - 2]; Table[ExpandAll[Ca[x, n]], {n, 0, 10}]; a = Table[CoefficientList[Ca[x, n], x], {n, 0, 10}]; Flatten[a]
Formula
Charlier polynomials: C_{n}(a; x) = Sum_{k=0..n} binomial(n,k)*binomial(x,k)*k!*(-a)^(n-k).
Extensions
Edited by Peter Luschny, Sep 21 2012
Comments