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

0,5

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 = 3.

Example

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, ...

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

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

Crossrefs

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

Sequence in context: A166455 A171142 A174037 * A131239 A114033 A334426

Adjacent sequences: A173074 A173075 A173076 * A173078 A173079 A173080

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 09 2010

Status

approved