OFFSET
0,4
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
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).
From G. C. Greubel, Jan 02 2022: (Start)
T(n, k) = 3^k * Sum_{j=0..k} binomial(n+1, j)*(-1)^j*(k-j+1)^n.
T(n, k) = 3^k * A008292(n, k+1).
T(n, 0) = 1.
T(n, n-1) = 3^n, for n >= 1. (End)
EXAMPLE
Triangle begins as:
1;
1;
1, 3;
1, 12, 9;
1, 33, 99, 27;
1, 78, 594, 702, 81;
1, 171, 2718, 8154, 4617, 243;
1, 360, 10719, 65232, 96471, 29160, 729;
1, 741, 38637, 421713, 1265139, 1043199, 180063, 2187;
1, 1506, 131472, 2382318, 12651390, 21440862, 10649232, 1097874, 6561;
MATHEMATICA
(* First program *)
p[x_, n_]:= (1-3*x)^(1+n)*PolyLog[-n, 3*x]/(3*x);
T[n_]:= CoefficientList[Series[p[x, n], {x, 0, 30}], x];
Table[T[n], {n, 0, 10}]//Flatten (* modified by G. C. Greubel, Jan 02 2022 *)
(* Second program *)
T[n_, k_]:= 3^k*Sum[(k-j+1)^n*Binomial[n+1, j]*(-1)^j, {j, 0, k}];
Join[{1}, Table[T[n, k], {n, 10}, {k, 0, n-1}]]//Flatten (* G. C. Greubel, Jan 02 2022 *)
PROG
(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]]) >;
[1] cat [T(n, k): k in [0..n-1], n in [0..10]]; // G. C. Greubel, Jan 02 2022
(Sage)
def T(n, k): return 3^k*sum((-1)^j*binomial(n+1, j)*(k-j+1)^n for j in (0..k))
[1]+flatten([[T(n, k) for k in (0..n-1)] for n in (0..10)]) # G. C. Greubel, Jan 02 2022
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Roger L. Bagula, Feb 08 2009
EXTENSIONS
Edited by G. C. Greubel, Jan 02 2022
STATUS
approved