A173077 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 = 3, read by rows.

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 12, 23, 12, 1, 1, 13, 36, 36, 13, 1, 1, 32, 122, 181, 122, 32, 1, 1, 33, 155, 304, 304, 155, 33, 1, 1, 88, 513, 1270, 1689, 1270, 513, 88, 1, 1, 89, 602, 1784, 2960, 2960, 1784, 602, 89, 1, 1, 252, 1988, 6923, 13817, 17261, 13817, 6923, 1988, 252, 1

Offset: 0

Author

Roger L. Bagula, Feb 09 2010

Keywords

• nonn
• tabl

Examples

			Triangle starts:
  1;
  1,   1;
  1,   4,    1;
  1,   5,    5,    1;
  1,  12,   23,   12,     1;
  1,  13,   36,   36,    13,     1;
  1,  32,  122,  181,   122,    32,     1;
  1,  33,  155,  304,   304,   155,    33,    1;
  1,  88,  513, 1270,  1689,  1270,   513,   88,    1;
  1,  89,  602, 1784,  2960,  2960,  1784,  602,   89,   1;
  1, 252, 1988, 6923, 13817, 17261, 13817, 6923, 1988, 252, 1;
  ...
Row sums: 1, 2, 6, 12, 49, 100, 491, 986, 5433, 10872, 63223, ...
		

Links

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

Crossrefs

Cf. A132044 (q=0), A173075 (q=1), A173076 (q=2), this sequence (q=3).

Programs

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,3): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021

Mathematica

T[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + 3^Floor[n/2] Binomial[n-2, k- 1]];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

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,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 09 2021

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 = 3.