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

1,5

Comments

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

Matrix inverse is A216916. - Peter Luschny, Sep 21 2012

Links

Table of n, a(n) for n=1..58.

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.

Formula

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

Example

[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.

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

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]

Crossrefs

Cf. A216916.

Sequence in context: A033789 A109532 A264584 * A176106 A302867 A058659

Adjacent sequences: A137335 A137336 A137337 * A137339 A137340 A137341

Keyword

tabl,sign

Author

Roger L. Bagula, Apr 07 2008

Extensions

Edited by Peter Luschny, Sep 21 2012

Status

approved