A173045 Triangle T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 3, read by rows.

1, 1, 1, 1, 10, 1, 1, 29, 29, 1, 1, 84, 6566, 84, 1, 1, 247, 14348916, 14348916, 247, 1, 1, 734, 282429536495, 150094635296999140, 282429536495, 734, 1, 1, 2193, 50031545098999727, 2503155504993241601315571986085883, 2503155504993241601315571986085883, 50031545098999727, 2193, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..11 of the triangle, flattened

Formula

T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 3.

Sum_{k=0..n} T(n, k, 3) = A000295(n) + Sum_{k=0..n} 3^(n*binomial(n-2, k-1)). - G. C. Greubel, Feb 19 2021

Example

Triangle begins as:
  1;
  1,   1;
  1,  10,            1;
  1,  29,           29,                  1;
  1,  84,         6566,                 84,            1;
  1, 247,     14348916,           14348916,          247,   1;
  1, 734, 282429536495, 150094635296999140, 282429536495, 734, 1;

Mathematica

T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + q^(n*Binomial[n-2, k-1])];
Table[t[n, k, 3], {n, 0, 9}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 19 2021 *)

Prog

(Sage)
def T(n, k, q):
    if (k==0 or k==n): return 1
    else: return binomial(n, k) -1 +q^(n*binomial(n-2, k-1))
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..9)]) # G. C. Greubel, Feb 19 2021
(Magma)
T:= func< n, k, q | k eq 0 or k eq n select 1 else Binomial(n, k) -1 +q^(n*Binomial(n-2, k-1)) >;
[T(n, k, 3): k in [0..n], n in [0..9]]; // G. C. Greubel, Feb 19 2021

Crossrefs

Cf. A132044 (q=0), A007318 (q=1), A173043 (q=2), this sequence (q=3).

Cf. A000295.

Sequence in context: A190152 A154984 A173047 * A176491 A008958 A168524

Adjacent sequences: A173042 A173043 A173044 * A173046 A173047 A173048

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 08 2010

Extensions

Edited by G. C. Greubel, Feb 19 2021

Status

approved