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%I #17 Sep 08 2022 08:45:41
%S 1,1,1,3,1,12,9,1,33,99,27,1,78,594,702,81,1,171,2718,8154,4617,243,1,
%T 360,10719,65232,96471,29160,729,1,741,38637,421713,1265139,1043199,
%U 180063,2187,1,1506,131472,2382318,12651390,21440862,10649232,1097874
%N Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.
%H G. C. Greubel, <a href="/A156366/b156366.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = [x^k]( p(n, x) ), where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*Sum_{j >= 0} ( (j+1)^n*(3*x)^j ), or p(n, x) = (1-3*x)^(n+1)* PolyLog(-n, 3*x)/(3*x).
%F From _G. C. Greubel_, Jan 02 2022: (Start)
%F T(n, k) = 3^k * Sum_{j=0..k} binomial(n+1, j)*(-1)^j*(k-j+1)^n.
%F T(n, k) = 3^k * A008292(n, k+1).
%F T(n, 0) = 1.
%F T(n, n-1) = 3^n, for n >= 1. (End)
%e Triangle begins as:
%e 1;
%e 1;
%e 1, 3;
%e 1, 12, 9;
%e 1, 33, 99, 27;
%e 1, 78, 594, 702, 81;
%e 1, 171, 2718, 8154, 4617, 243;
%e 1, 360, 10719, 65232, 96471, 29160, 729;
%e 1, 741, 38637, 421713, 1265139, 1043199, 180063, 2187;
%e 1, 1506, 131472, 2382318, 12651390, 21440862, 10649232, 1097874, 6561;
%t (* First program *)
%t p[x_, n_]:= (1-3*x)^(1+n)*PolyLog[-n, 3*x]/(3*x);
%t T[n_]:= CoefficientList[Series[p[x, n], {x,0,30}], x];
%t Table[T[n], {n,0,10}]//Flatten (* modified by _G. C. Greubel_, Jan 02 2022 *)
%t (* Second program *)
%t T[n_, k_]:= 3^k*Sum[(k-j+1)^n*Binomial[n+1, j]*(-1)^j, {j,0,k}];
%t Join[{1}, Table[T[n, k], {n, 10}, {k, 0, n-1}]]//Flatten (* _G. C. Greubel_, Jan 02 2022 *)
%o (Magma) T:= func< n,k | n eq 0 and k eq 0 select 1 else 3^k*(&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k]]) >;
%o [1] cat [T(n,k): k in [0..n-1], n in [0..10]]; // _G. C. Greubel_, Jan 02 2022
%o (Sage)
%o def T(n, k): return 3^k*sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k))
%o [1]+flatten([[T(n,k) for k in (0..n-1)] for n in (0..10)]) # _G. C. Greubel_, Jan 02 2022
%Y Cf. A000244, A008292.
%K nonn,tabf
%O 0,4
%A _Roger L. Bagula_, Feb 08 2009
%E Edited by _G. C. Greubel_, Jan 02 2022
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