A174719 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 3, read by rows.

1, 1, 1, 1, -7, 1, 1, -51, -51, 1, 1, -239, -399, -239, 1, 1, -967, -2177, -2177, -967, 1, 1, -3639, -10191, -13831, -10191, -3639, 1, 1, -13115, -43719, -74323, -74323, -43719, -13115, 1, 1, -45919, -177119, -360799, -452639, -360799, -177119, -45919, 1

Offset

0,5

Comments

The row sums of this class of sequences, for varying q, is given by Sum_{k=0..n} T(n, k, q) = q^n * (n+1) + 2^n * (1 - q^n). - G. C. Greubel, Feb 09 2021

Links

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

Formula

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

Sum_{k=0..n} T(n, k, 3) = 3^n*(n+1) + 2^n*(1 - 3^n) = A027471(n+2) - A248216(n). - G. C. Greubel, Feb 09 2021

Example

Triangle begins as:
  1;
  1,      1;
  1,     -7,       1;
  1,    -51,     -51,       1;
  1,   -239,    -399,    -239,       1;
  1,   -967,   -2177,   -2177,    -967,       1;
  1,  -3639,  -10191,  -13831,  -10191,   -3639,       1;
  1, -13115,  -43719,  -74323,  -74323,  -43719,  -13115,      1;
  1, -45919, -177119, -360799, -452639, -360799, -177119, -45919, 1;

Mathematica

T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1);
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Sage)
def T(n, k, q): return 1 + (1-q^n)*(binomial(n, k) - 1)
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 09 2021
(Magma)
T:= func< n, k, q | 1 + (1-q^n)*(Binomial(n, k) -1) >;
[T(n, k, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 09 2021

Crossrefs

Cf. A000012 (q=1), A174718 (q=2), this sequence (q=3), A174720 (q=4).

Cf. A027471, A248216.

Sequence in context: A203389 A174689 A172345 * A176392 A015118 A340428

Adjacent sequences: A174716 A174717 A174718 * A174720 A174721 A174722

Keyword

sign,tabl

Author

Roger L. Bagula, Mar 28 2010

Extensions

Edited by G. C. Greubel, Feb 09 2021

Status

approved