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%I #14 Oct 28 2019 04:28:46
%S 1,3,1,9,7,1,27,38,12,1,81,192,101,18,1,243,969,755,215,25,1,729,5115,
%T 5494,2205,400,33,1,2187,29322,40971,21469,5355,679,42,1,6561,187992,
%U 323658,209356,66619,11452,1078,52,1,19683,1370745,2764926,2111318,813645,176295,22302,1626,63,1
%N Coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3). Triangle read by rows, T(n, k) for 0 <= k <= n.
%C 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 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 T(n, k) = n!*[x^k] p(n) where p(n) = [t^n] exp(3*t)*(1-t)^(-x).
%e The triangle starts:
%e {1}
%e {3, 1}
%e {9, 7, 1}
%e {27, 38, 12, 1}
%e {81, 192, 101, 18, 1}
%e {243, 969, 755, 215, 25, 1}
%e {729, 5115, 5494, 2205, 400, 33, 1}
%e {2187, 29322, 40971, 21469, 5355, 679, 42, 1}
%e {6561, 187992, 323658, 209356, 66619, 11452, 1078, 52, 1}
%e {19683, 1370745, 2764926, 2111318, 813645, 176295, 22302, 1626, 63, 1}
%p egf := exp(3*t)*(1-t)^(-x): ser := series(egf, t, 12): p := n -> coeff(ser, t, n):
%p seq(print(n!*seq(coeff(p(n), x, k), k=0..n)), n=0..9);
%t p [n_] := HypergeometricU[-n, 1 - n - x, 3];
%t Table[CoefficientList[p[n], x], {n, 0, 9}] // Flatten
%Y A094816 (z=1), |A137346| (z=2), this sequence (z=3).
%Y Row sums in A053486.
%K nonn,tabl
%O 0,2
%A _Peter Luschny_, Oct 27 2019
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