A137346 Coefficients of a special case of Poisson-Charlier polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.

1, -2, 1, 4, -5, 1, -8, 20, -9, 1, 16, -78, 59, -14, 1, -32, 324, -360, 135, -20, 1, 64, -1520, 2254, -1165, 265, -27, 1, -128, 8336, -15232, 9954, -3045, 469, -35, 1, 256, -53872, 113868, -88508, 33649, -6888, 770, -44, 1, -512, 405600, -948840, 839684, -376278, 95025, -14028, 1194, -54

Offset: 0

Author

Roger L. Bagula, Apr 08 2008

Keywords

• tabl
• sign

Examples

			{1},
{-2, 1},
{4, -5, 1},
{-8, 20, -9, 1},
{16, -78,59, -14, 1},
{-32, 324, -360, 135, -20, 1},
{64, -1520, 2254, -1165, 265, -27, 1},
{-128, 8336, -15232, 9954, -3045, 469, -35, 1},
{256, -53872, 113868, -88508, 33649, -6888, 770, -44, 1},
{-512, 405600, -948840, 839684, -376278, 95025, -14028, 1194, -54, 1},
{1024, -3492416, 8793216, -8592220,4373060, -1297569, 235473, -26370, 1770, -65, 1}
		

Links

• M. Dunster, Uniform asymptotic expansions for Charlier polynomials, J. Approx. Theory, 112 (2001) pp. 93-133.
• Eric Weisstein's World of Mathematics, Poisson-Charlier Polynomial.

Crossrefs

Cf. A046716, A092553, A094816, A159964.

Programs

Maple

R := proc(n) add((-1)^k*binomial(n,k)* k!*2^(n-k)*binomial(-x, k), k=0..n);
expand(%) end: p := n -> seq((-1)^(n-k)*coeff(R(n), x, k), k=0..n):
seq(p(n), n = 0..9);
# Or:
egf := exp(-2*t)*(1+t)^x: ser := series(egf, t, 12): p := n -> coeff(ser, t, n):
seq(n!*seq(coeff(p(n), x, k), k=0..n), n=0..9); # Peter Luschny, Oct 27 2019

Mathematica

Ca[x, 0] = 1; Ca[x, 1] = -2 + x;
Ca[x_, n_] := Ca[x, n] = (x - n - 1) Ca[x, n - 1] - 2 (n - 1) Ca[x, n - 2];
Table[CoefficientList[Ca[x, n], x], {n, 0, 9}] // Flatten
(* The unsigned row polynomials (see Peter Bala's comment) are: *)
R[n_] := HypergeometricU[-n, 1 - n - x, 2];
Table[R[n], {n, 0, 6}] (* Peter Luschny, Oct 27 2019 *)

Formula

T(n, k) = n!*[x^k] p(n) where p(n) = [t^n] exp(-2*t)*(1+t)^x.

With p(0, x) = 1 and p(1, x) = x - 2 the polynomials obey the recurrence

p(n, x) = (x - n - 1)*p(n-1, x) - 2*(n - 1)*p(n-2, x).

Row sums are (-2)^n*(n-1) = (-1)^n*A159964(n-1).

From Peter Bala, Oct 23 2019: (Start)

The unsigned row polynomials are

R(n,x) = Sum_{k=0..n} (-1)^k*binomial(n, k)*k!*2^(n-k)*binomial(-x, k).

They occur in series acceleration formulas for the constant

1/e^2 = n!*2^n*Sum_{k >= 0}(-2)^k/(k!*R(n,k)*R(n,k+1)) = 0.1353 35283 23661 ... (cf. A092553, A046716, A094816).

(End)

R(n, x) = KummerU(-n, 1 - n - x, 2). - Peter Luschny, Oct 27 2019

Extensions

Edited by Peter Luschny, Oct 27 2019