login
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.
3

%I #33 Oct 28 2019 02:26:12

%S 1,-1,1,0,-3,1,3,6,-6,1,-12,-9,26,-10,1,45,3,-109,71,-15,1,-198,81,

%T 501,-475,155,-21,1,1071,-786,-2663,3329,-1455,295,-28,1,-6984,6711,

%U 16510,-25495,13729,-3647,511,-36,1,53217,-60309,-117912,216004,-135961,43897,-7994,826,-45,1,-462330,589197,953711

%N 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.

%C Row sums are 1, 0, -2, 4, -4, -4, 44, -236, 1300, -8276, 61484, etc.

%C Matrix inverse is A216916. - _Peter Luschny_, Sep 21 2012

%H Carl V. L. Charlier, <a href="https://babel.hathitrust.org/cgi/pt?id=chi.73230316&amp;view=1up&amp;seq=103">Über die Darstellung willkürlicher Funktionen</a>, 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 <a href="https://www.hathitrust.org/accessibility">HathiTrust Digital Library</a>.]

%H M. Dunster, <a href="https://doi.org/10.1006/jath.2001.3595"> Uniform asymptotic expansions for Charlier polynomials</a>, J. Approx. Theory, 112 (2001), pp. 93-133.

%H Chun-Hua Ou, <a href="http://math.nist.gov/~DLozier/SF21/SF21slides/Ou.pdf">Global Asymptotics of the Charlier Polynomials via the Riemann-Hilbert Method</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Carl_Charlier">Carl Charlier</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Charlier_polynomials">Charlier polynomials</a>.

%F Charlier polynomials: C_{n}(a; x) = Sum_{k=0..n} binomial(n,k)*binomial(x,k)*k!*(-a)^(n-k).

%e [0] 1,

%e [1] -1, 1,

%e [2] 0, -3, 1,

%e [3] 3, 6, -6, 1,

%e [4] -12, -9, 26, -10, 1,

%e [5] 45, 3, -109, 71, -15, 1,

%e [6] -198, 81, 501, -475, 155, -21, 1,

%e [7] 1071, -786, -2663, 3329, -1455, 295, -28, 1,

%e [8] -6984, 6711, 16510, -25495, 13729, -3647, 511, -36, 1,

%e [9] 53217, -60309, -117912, 216004, -135961, 43897, -7994, 826, -45, 1.

%p with(PolynomialTools):

%p C := (n, x) -> if n>0 then expand((x-n)*C(n-1,x)-n*C(n-2,x))

%p elif n = 0 then 1 else 0 fi:

%p A137338_row := n -> CoefficientList(C(n,x), x);

%p for n from 0 to 7 do A137338_row(n) od;

%p # _Peter Luschny_, Sep 21 2012

%t 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]

%Y Cf. A216916.

%K tabl,sign

%O 1,5

%A _Roger L. Bagula_, Apr 07 2008

%E Edited by _Peter Luschny_, Sep 21 2012