A174720 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 4, read by rows.

1, 1, 1, 1, -14, 1, 1, -125, -125, 1, 1, -764, -1274, -764, 1, 1, -4091, -9206, -9206, -4091, 1, 1, -20474, -57329, -77804, -57329, -20474, 1, 1, -98297, -327659, -557021, -557021, -327659, -98297, 1, 1, -458744, -1769444, -3604424, -4521914, -3604424, -1769444, -458744, 1

Offset: 0

Author

Roger L. Bagula, Mar 28 2010

Keywords

• sign
• tabl

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

Examples

			Triangle begins as:
  1;
  1,       1;
  1,     -14,        1;
  1,    -125,     -125,        1;
  1,    -764,    -1274,     -764,        1;
  1,   -4091,    -9206,    -9206,    -4091,        1;
  1,  -20474,   -57329,   -77804,   -57329,   -20474,        1;
  1,  -98297,  -327659,  -557021,  -557021,  -327659,   -98297,       1;
  1, -458744, -1769444, -3604424, -4521914, -3604424, -1769444, -458744, 1;
		

Links

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

Crossrefs

Cf. A000012 (q=1), A174718 (q=2), A174719 (q=3), this sequence (q=4).

Cf. A002697, A248217.

Programs

Magma

T:= func< n,k,q | 1 + (1-q^n)*(Binomial(n,k) -1) >;
[T(n,k,4): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 09 2021

Mathematica

T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1);
Table[T[n,k,4], {n,0,12}, {k,0,n}]//Flatten

Sage

def T(n,k,q): return 1 + (1-q^n)*(binomial(n,k) - 1)
flatten([[T(n,k,4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 09 2021

Formula

T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q=4.

Sum_{k=0..n} T(n, k, 4) = 4^n*(n+1) + 2^n*(1 - 4^n) = A002697(n+1) - A248217(n). - G. C. Greubel, Feb 09 2021

Extensions

Edited by G. C. Greubel, Feb 09 2021