A173043 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 = 2, read by rows.

1, 1, 1, 1, 5, 1, 1, 10, 10, 1, 1, 19, 261, 19, 1, 1, 36, 32777, 32777, 36, 1, 1, 69, 16777230, 68719476755, 16777230, 69, 1, 1, 134, 34359738388, 1180591620717411303458, 1180591620717411303458, 34359738388, 134, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..12 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 = 2.

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

Example

Triangle begins as:
  1;
  1,  1;
  1,  5,        1;
  1, 10,       10,           1;
  1, 19,      261,          19,        1;
  1, 36,    32777,       32777,       36,  1;
  1, 69, 16777230, 68719476755, 16777230, 69, 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, 2], {n, 0, 12}, {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, 2) for k in (0..n)] for n in (0..12)]) # 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, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2021

Crossrefs

Cf. A132044 (q=0), A007318 (q=1), this sequence (q=2), A173045 (q=3).

Cf. A000295.

Sequence in context: A188461 A188474 A173046 * A082046 A132787 A181370

Adjacent sequences: A173040 A173041 A173042 * A173044 A173045 A173046

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 08 2010

Extensions

Edited by G. C. Greubel, Feb 19 2021

Status

approved