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