A173047 Triangle T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 3, read by rows.

1, 1, 1, 1, 10, 1, 1, 29, 29, 1, 1, 84, 167, 84, 1, 1, 247, 738, 738, 247, 1, 1, 734, 2930, 4393, 2930, 734, 1, 1, 2193, 10955, 21904, 21904, 10955, 2193, 1, 1, 6568, 39393, 98470, 131289, 98470, 39393, 6568, 1, 1, 19691, 137816, 413426, 689030, 689030, 413426, 137816, 19691, 1

Offset

0,5

Comments

The triangle sequences having the form T(n,k,q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 have the row sums Sum_{k=0..n} T(n,k,q) = 2^(n-2)*q^n + 2^n - (n-1) - (5/4)*[n=0] -(q/2)*[n=1]. - G. C. Greubel, Feb 16 2021

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Formula

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

Sum_{k=0..n} T(n, k, 3) = (1/4)*(6^n + 2^(n+2) - 4*(n-1) - 5*[n=0] - 6*[n=1]). - G. C. Greubel, Feb 16 2021

Example

Ttiangle begins as:
  1;
  1,     1;
  1,    10,      1;
  1,    29,     29,      1;
  1,    84,    167,     84,      1;
  1,   247,    738,    738,    247,      1;
  1,   734,   2930,   4393,   2930,    734,      1;
  1,  2193,  10955,  21904,  21904,  10955,   2193,      1;
  1,  6568,  39393,  98470, 131289,  98470,  39393,   6568,     1;
  1, 19691, 137816, 413426, 689030, 689030, 413426, 137816, 19691, 1;

Mathematica

T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] +(q^n)*Binomial[n-2, k-1] -1];
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 16 2021 *)

Prog

(Sage)
def T(n, k, q): return 1 if (k==0 or k==n) else binomial(n, k) + q^n*binomial(n-2, k-1) -1
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 16 2021
(Magma)
T:= func< n, k, q | k eq 0 or k eq n select 1 else Binomial(n, k) + q^n*Binomial(n-2, k-1) -1 >;
[T(n, k, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021

Crossrefs

Cf. A132044 (q=0), A173075 (q=1), A173046 (q=2), this sequence (q=3).

Sequence in context: A146765 A190152 A154984 * A173045 A176491 A008958

Adjacent sequences: A173044 A173045 A173046 * A173048 A173049 A173050

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 08 2010

Extensions

Edited by G. C. Greubel, Feb 16 2021

Status

approved