A173007 Triangle T(n,k) read by rows: coefficient [x^k] of the polynomial Product_{i=1..n} (x + q^i) in row n and q = 3.

1, 3, 1, 27, 12, 1, 729, 351, 39, 1, 59049, 29160, 3510, 120, 1, 14348907, 7144929, 882090, 32670, 363, 1, 10460353203, 5223002148, 650188539, 24698520, 297297, 1092, 1, 22876792454961, 11433166050879, 1427185336941, 54665851779, 674887059, 2685501, 3279, 1

Offset

0,2

Comments

Triangle T(n,k), read by rows, given by [3,6,27,72,243,702,2187,6480,...] DELTA [1,0,3,0,9,0,27,0,81,0,243,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 01 2011

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

p(x,n,q) = 1 if n=0, Product_{i=1..n} (x + q^i) otherwise, with q=3.

T(n,k) = 3^n*T(n-1,k) + T(n-1,k-1), T(0,0)=1. - Philippe Deléham, Oct 01 2011

Sum_{k=0..n} T(n, k, 4) = A290000(n+1). - G. C. Greubel, Feb 20 2021

Example

Triangle begins as:
            1;
            3,          1;
           27,         12,         1;
          729,        351,        39,        1;
        59049,      29160,      3510,      120,      1;
     14348907,    7144929,    882090,    32670,    363,    1;
  10460353203, 5223002148, 650188539, 24698520, 297297, 1092, 1;

Mathematica

(* First program *)
p[x_, n_, q_] = If[n==0, 1, Product[x + q^i, {i, 1, n}]];
Table[CoefficientList[p[x, n, 3], x], {n, 0, 10}] (* modified by G. C. Greubel, Feb 20 2021 *)
(* Second program *)
T[n_, k_, q_]:= If[k<0 || k>n, 0, If[k==n, 1, q^n*T[n-1, k, q] +T[n-1, k-1, q] ]];
Table[T[n, k, 3], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 20 2021 *)

Prog

(Sage)
def T(n, k, q):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    else: return q^n*T(n-1, k, q) + T(n-1, k-1, q)
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 20 2021
(Magma)
function T(n, k, q)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  else return q^n*T(n-1, k, q) + T(n-1, k-1, q);
  end if; return T; end function;
[T(n, k, 3): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 20 2021

Crossrefs

Cf. A023531 (q=0), A007318 (q=1), A108084 (q=2), this sequence (q=3), A173008 (q=4).

Cf. A203148, A290000.

Sequence in context: A289329 A033464 A170924 * A113099 A317930 A270078

Adjacent sequences: A173004 A173005 A173006 * A173008 A173009 A173010

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 07 2010

Extensions

Edited by G. C. Greubel, Feb 20 2021

Status

approved