OFFSET
0,5
COMMENTS
This sequence belongs to the class defined by T(n, m, q) = 2*T(n, m, q-1) - 1. The first few q values gives the sequences: A007318 (q=0), A109128 (q=1), A131061 (q=2), A168625 (q=3), this sequence (q=4).
Row sums are: {1, 2, 19, 68, 181, 422, 919, 1928, 3961, 8042, 16219, ...}.
Former title: A recursive symmetrical triangular sequence:q=4: t(n, m, q) = 2*t(n, m, q-1) - 1. - G. C. Greubel, Mar 12 2020
LINKS
G. C. Greubel, Rows n = 0..100 of the triangle, flattened
FORMULA
T(n, m, q) = 2*T(n, m, q-1) - 1, with T(n, m, 0) = binomial(n, m) and q = 4.
From G. C. Greubel, Mar 12 2020: (Start)
T(n, k, q) = 2^q * binomial(n, k) - (2^q - 1), with q = 4.
Sum_{k=0..n} T(n, k, q) = 2^(n + q) - (n + 1)*(2^q - 1) (row sums). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 17, 1;
1, 33, 33, 1;
1, 49, 81, 49, 1;
1, 65, 145, 145, 65, 1;
1, 81, 225, 305, 225, 81, 1;
1, 97, 321, 545, 545, 321, 97, 1;
1, 113, 433, 881, 1105, 881, 433, 113, 1;
1, 129, 561, 1329, 2001, 2001, 1329, 561, 129, 1;
1, 145, 705, 1905, 3345, 4017, 3345, 1905, 705, 145, 1;
MAPLE
A176203:= (n, k) -> 16*binomial(n, k) -15; seq(seq(A176203(n, k), k = 0..n), n = 0.. 12); # G. C. Greubel, Mar 12 2020
MATHEMATICA
T[n_, m_, q]:= 2^q*(Binomial[n, m] -1) + 1; Table[T[n, m, 4], {n, 0, 12}, {m, 0, n} ]//Flatten (* modified by G. C. Greubel, Mar 12 2020 *)
Table[16*Binomial[n, k] -15, {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)
PROG
(Magma) [16*Binomial(n, k) -15: k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 12 2020
(Sage) [[16*binomial(n, k) -15 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Mar 12 2020
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Apr 11 2010
EXTENSIONS
Edited by G. C. Greubel, Mar 12 2020
STATUS
approved