A156366 - OEIS

%I #20 Mar 19 2026 18:13:55

%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 (* Alternative: *)

%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 (SageMath)

%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