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=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
EXAMPLE
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;
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
PROG
(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
(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
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Mar 28 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 09 2021
STATUS
approved