A327997 Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).

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

Author

Peter Luschny, Oct 27 2019

Keywords

• nonn
• tabl

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.

Examples

			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;
		

Crossrefs

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

Columns k=0..3 give A000244, A346395, A381052, A382701.

Row sums in A053486.

Programs

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

PARI

T(n, k) = sum(j=k, n, 3^(n-j)*binomial(n, j)*abs(stirling(j, k, 1))); \\ Seiichi Manyama, Apr 19 2025

Formula

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

From Igor Victorovich Statsenko, Feb 14 2025: (Start)

T(m, n, k) = Sum_{i=0..n} Stirling1(n-i, k)*binomial(n, i)*m^(i)*(-1)^(n-k), for m = -3.

The triangle T(n,k) is a representative of the parametric family of triangles T(m,n,k), whose columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)*exp(-m*x)/k! for the case m=-3. (End)