A327997 Coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3). Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 3, 1, 9, 7, 1, 27, 38, 12, 1, 81, 192, 101, 18, 1, 243, 969, 755, 215, 25, 1, 729, 5115, 5494, 2205, 400, 33, 1, 2187, 29322, 40971, 21469, 5355, 679, 42, 1, 6561, 187992, 323658, 209356, 66619, 11452, 1078, 52, 1, 19683, 1370745, 2764926, 2111318, 813645, 176295, 22302, 1626, 63, 1

Offset

0,2

Comments

KummerU(-n, 1-n-x, 1) are the Charlier polynomials with coefficients in A094816, the coefficients of KummerU(-n, 1-n-x, 2) are in |A137346|.

The exponential generating function of this family of sequences of polynomials is in its general form (1-t)^(-x)*exp(alpha*t) with a parameter alpha.

Links

Table of n, a(n) for n=0..54.

Formula

T(n, k) = n!*[x^k] p(n) where p(n) = [t^n] exp(3*t)*(1-t)^(-x).

Example

The triangle starts:
{1}
{3,     1}
{9,     7,       1}
{27,    38,      12,      1}
{81,    192,     101,     18,      1}
{243,   969,     755,     215,     25,     1}
{729,   5115,    5494,    2205,    400,    33,     1}
{2187,  29322,   40971,   21469,   5355,   679,    42,    1}
{6561,  187992,  323658,  209356,  66619,  11452,  1078,  52,   1}
{19683, 1370745, 2764926, 2111318, 813645, 176295, 22302, 1626, 63, 1}

Maple

egf := exp(3*t)*(1-t)^(-x): ser := series(egf, t, 12): p := n -> coeff(ser, t, n):
seq(print(n!*seq(coeff(p(n), x, k), k=0..n)), n=0..9);

Mathematica

p [n_] := HypergeometricU[-n, 1 - n - x, 3];
Table[CoefficientList[p[n], x], {n, 0, 9}] // Flatten

Crossrefs

A094816 (z=1), |A137346| (z=2), this sequence (z=3).

Row sums in A053486.

Sequence in context: A197730 A231902 A143495 * A245789 A210395 A019770

Adjacent sequences: A327994 A327995 A327996 * A327998 A327999 A328000

Keyword

nonn,tabl

Author

Peter Luschny, Oct 27 2019

Status

approved