A173076 Triangle T(n, k, q) = binomial(n, k) - 1 + q^(floor(n/2))*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, 3, 1, 1, 4, 4, 1, 1, 7, 13, 7, 1, 1, 8, 21, 21, 8, 1, 1, 13, 46, 67, 46, 13, 1, 1, 14, 60, 114, 114, 60, 14, 1, 1, 23, 123, 295, 389, 295, 123, 23, 1, 1, 24, 147, 419, 685, 685, 419, 147, 24, 1, 1, 41, 300, 1015, 2001, 2491, 2001, 1015, 300, 41, 1

Offset

0,5

Links

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

Formula

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

Example

Triangle begins as:
  1;
  1,  1;
  1,  3,   1;
  1,  4,   4,    1;
  1,  7,  13,    7,    1;
  1,  8,  21,   21,    8,    1;
  1, 13,  46,   67,   46,   13,    1;
  1, 14,  60,  114,  114,   60,   14,    1;
  1, 23, 123,  295,  389,  295,  123,   23,   1;
  1, 24, 147,  419,  685,  685,  419,  147,  24,  1;
  1, 41, 300, 1015, 2001, 2491, 2001, 1015, 300, 41, 1;

Mathematica

T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + q^(Floor[n/2])*Binomial[n-2, k-1]];
Table[T[n, k, 2], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(Magma)
T:= func< n, k, q | k eq 0 or k eq n select 1 else Binomial(n, k) + q^(Floor(n/2))*Binomial(n-2, k-1) -1 >;
[T(n, k, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021
(Sage)
def T(n, k, q): return 1 if (k==0 or k==n) else binomial(n, k) + q^(n//2)*binomial(n-2, k-1) -1
flatten([[T(n, k, 1) for k in (0..n)] for n in (0..12)])

Crossrefs

Cf. A132044 (q=0), A173075 (q=1), this sequence (q=2), A173077 (q=3).

Sequence in context: A174032 A180979 A102716 * A134510 A335322 A171145

Adjacent sequences: A173073 A173074 A173075 * A173077 A173078 A173079

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 09 2010

Extensions

Edited by G. C. Greubel, Jul 09 2021

Status

approved