A172198 Triangle T(n, k, q) = 1 + abs(c(n,q) - c(k,q))*abs(c(n,q) - c(n-k, q)), where c(n,q) = Product_{j=1..n} (1 - q^j) and q = 2, read by rows.

1, 1, 1, 1, 325, 1, 1, 178849, 178849, 1, 1, 1121470273, 1106493697, 1121470273, 1, 1, 65131063096321, 64859828626945, 64859828626945, 65131063096321, 1, 1, 34423599076368353281, 34376183545107456001, 34376383642256188417, 34376183545107456001, 34423599076368353281, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Formula

T(n, k, q) = 1 + abs(c(n,q) - c(k,q))*abs(c(n,q) - c(n-k, q)), where c(n,q) = Product_{j=1..n} (1 - q^j) and q = 3.

Example

Triangle begins as:
  1;
  1,              1;
  1,            325,              1;
  1,         178849,         178849,              1;
  1,     1121470273,     1106493697,     1121470273,              1;
  1, 65131063096321, 64859828626945, 64859828626945, 65131063096321, 1;

Mathematica

T[n_, k_, q_]:= 1 + Abs[QPochhammer[q, q, n] -QPochhammer[q, q, k]]*Abs[QPochhammer[q, q, n] - QPochhammer[q, q, n-k]];
Table[T[n, k, 3], {n, 0, 10}, {k, 0, n}]//TableForm (* modified by G. C. Greubel, May 06 2021 *)

Prog

(Magma)
c:= func< n, q | n eq 0 select 1 else (&*[1-q^j: j in [1..n]]) >;
T:= func< n, k, q | 1 + Abs(c(n, q) - c(k, q))*Abs(c(n, q) - c(n-k, q)) >;
[T(n, k, 3): k in [0..n], n in [0..10]]; // G. C. Greubel, May 06 2021
(Sage)
from sage.combinat.q_analogues import q_pochhammer
def T(n, k, q): return 1 + abs(q_pochhammer(n, q, q) -q_pochhammer(k, q, q))*abs(q_pochhammer(n, q, q) -q_pochhammer(n-k, q, q))
[[T(n, k, 3) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 06 2021

Crossrefs

Cf. A172196 (q=2), this sequence (q=3).

Sequence in context: A202635 A013763 A013887 * A343080 A159976 A253433

Adjacent sequences: A172195 A172196 A172197 * A172199 A172200 A172201

Keyword

nonn,tabl,less,easy

Author

Roger L. Bagula, Jan 29 2010

Extensions

Edited by G. C. Greubel, May 06 2021

Status

approved