A174389 Triangle T(n, k) = c(n, q)/c(k, q) if k <= floor(n/2), otherwise c(n, q)/c(n-k, q), where c(n, q) = Product_{j=1..n} (1 - q^j) and q = 4, read by rows.

1, 1, 1, 1, -15, 1, 1, -63, -63, 1, 1, -255, 16065, -255, 1, 1, -1023, 260865, 260865, -1023, 1, 1, -4095, 4189185, -1068242175, 4189185, -4095, 1, 1, -16383, 67088385, -68631417855, -68631417855, 67088385, -16383, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = c(n, q)/c(k, q) if k <= floor(n/2), otherwise c(n, q)/c(n-k, q), where c(n, q) = Product_{j=1..n} (1 - q^j) and q = 4.

T(n, n-k) = T(n, k).

Example

Triangle begins as:
  1;
  1,      1;
  1,    -15,        1;
  1,    -63,      -63,            1;
  1,   -255,    16065,         -255,            1;
  1,  -1023,   260865,       260865,        -1023,        1;
  1,  -4095,  4189185,  -1068242175,      4189185,    -4095,      1;
  1, -16383, 67088385, -68631417855, -68631417855, 67088385, -16383,     1;

Mathematica

c[n_, q_]= Product[1-q^i, {i, n}];
T[n_, k_, q_]= If[Floor[n/2]>=k, c[n, q]/c[n-k, q], c[n, q]/c[k, q]];
Table[T[n, k, 4], {n, 0, 12}, {k, 0, n}]//Flatten
(* Second program *)
T[n_, k_, q_]= With[{QP=QPochhammer}, If[k<=Floor[n/2], QP[q, q, n]/QP[q, q, n-k], QP[q, q, n]/QP[q, q, k]]];
Table[T[n, k, 4], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 04 2022 *)

Prog

(Magma)
QPochhammer:= func< n, a, q | n eq 0 select 1 else (&*[1-a*q^j: j in [0..n-1]]) >;
T:= func< n, k, q | k le Floor(n/2) select QPochhammer(n, q, q)/QPochhammer(n-k, q, q) else QPochhammer(n, q, q)/QPochhammer(k, q, q) >;
A174389:= func< n, k | T(n, k, 4) >;
[A174389(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 04 2022
(SageMath)
from sage.combinat.q_analogues import q_pochhammer
def T(n, k, q):
    if ((n//2)>k-1): return q_pochhammer(n, q, q)/q_pochhammer(n-k, q, q)
    else: return q_pochhammer(n, q, q)/q_pochhammer(k, q, q)
def A174389(n, k): return T(n, k, 4)
flatten([[A174389(n, k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Dec 04 2022

Crossrefs

Triangles of q: A000012 (q=0), A174387 (q=2), A174388 (q=3), this sequence (q=4).

Sequence in context: A172429 A040225 A070644 * A176286 A111805 A238754

Adjacent sequences: A174386 A174387 A174388 * A174390 A174391 A174392

Keyword

sign,tabl

Author

Roger L. Bagula, Mar 18 2010

Extensions

Edited by G. C. Greubel, Dec 04 2022

Status

approved