A156366 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.

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

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

Cf. A000244, A008292.

Sequence in context: A143492 A243662 A062139 * A144353 A356146 A039811

Adjacent sequences: A156363 A156364 A156365 * A156367 A156368 A156369

Keyword

nonn,tabf

Author

Roger L. Bagula, Feb 08 2009

Extensions

Edited by G. C. Greubel, Jan 02 2022

Status

approved