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

%I A327997 #35 Apr 19 2025 14:36:16
%S A327997 1,3,1,9,7,1,27,38,12,1,81,192,101,18,1,243,969,755,215,25,1,729,5115,
%T A327997 5494,2205,400,33,1,2187,29322,40971,21469,5355,679,42,1,6561,187992,
%U A327997 323658,209356,66619,11452,1078,52,1,19683,1370745,2764926,2111318,813645,176295,22302,1626,63,1
%N A327997 Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).
%C A327997 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|.
%C A327997 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.
%F A327997 T(n, k) = n!*[x^k] p(n) where p(n) = [t^n] exp(3*t)*(1-t)^(-x).
%F A327997 From _Igor Victorovich Statsenko_, Feb 14 2025: (Start)
%F A327997 T(m, n, k) = Sum_{i=0..n} Stirling1(n-i, k)*binomial(n, i)*m^(i)*(-1)^(n-k), for m = -3.
%F A327997 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)
%e A327997 The triangle starts:
%e A327997       1;
%e A327997       3,       1;
%e A327997       9,       7,       1;
%e A327997      27,      38,      12,       1;
%e A327997      81,     192,     101,      18,      1;
%e A327997     243,     969,     755,     215,     25,      1;
%e A327997     729,    5115,    5494,    2205,    400,     33,     1;
%e A327997    2187,   29322,   40971,   21469,   5355,    679,    42,    1;
%e A327997    6561,  187992,  323658,  209356,  66619,  11452,  1078,   52,  1;
%e A327997   19683, 1370745, 2764926, 2111318, 813645, 176295, 22302, 1626, 63, 1;
%p A327997 egf := exp(3*t)*(1-t)^(-x): ser := series(egf, t, 12): p := n -> coeff(ser, t, n):
%p A327997 seq(print(n!*seq(coeff(p(n), x, k), k=0..n)), n=0..9);
%t A327997 p [n_] := HypergeometricU[-n, 1 - n - x, 3];
%t A327997 Table[CoefficientList[p[n], x], {n, 0, 9}] // Flatten
%o A327997 (PARI) T(n, k) = sum(j=k, n, 3^(n-j)*binomial(n, j)*abs(stirling(j, k, 1))); \\ _Seiichi Manyama_, Apr 19 2025
%Y A327997 A094816 (z=1), |A137346| (z=2), this sequence (z=3).
%Y A327997 Columns k=0..3 give A000244, A346395, A381052, A382701.
%Y A327997 Row sums in A053486.
%K A327997 nonn,tabl
%O A327997 0,2
%A A327997 _Peter Luschny_, Oct 27 2019