A174387 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 = 2, read by rows.

1, 1, 1, 1, -3, 1, 1, -7, -7, 1, 1, -15, 105, -15, 1, 1, -31, 465, 465, -31, 1, 1, -63, 1953, -29295, 1953, -63, 1, 1, -127, 8001, -248031, -248031, 8001, -127, 1, 1, -255, 32385, -2040255, 63247905, -2040255, 32385, -255, 1, 1, -511, 130305, -16548735, 1042570305, 1042570305, -16548735, 130305, -511, 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 = 2.

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

Example

Triangle begins as:
  1;
  1,    1;
  1,   -3,      1;
  1,   -7,     -7,         1;
  1,  -15,    105,       -15,          1;
  1,  -31,    465,       465,        -31,          1;
  1,  -63,   1953,    -29295,       1953,        -63,         1;
  1, -127,   8001,   -248031,    -248031,       8001,      -127,      1;
  1, -255,  32385,  -2040255,   63247905,   -2040255,     32385,   -255,    1;
  1, -511, 130305, -16548735, 1042570305, 1042570305, -16548735, 130305, -511, 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, 2], {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, 2], {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) >;
A174387:= func< n, k | T(n, k, 2) >;
[A174387(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 A174387(n, k): return T(n, k, 2)
flatten([[A174387(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), this sequence (q=2), A174388 (q=3), A174389 (q=4).

Sequence in context: A058669 A057004 A059328 * A176791 A259471 A220555

Adjacent sequences: A174384 A174385 A174386 * A174388 A174389 A174390

Keyword

sign,tabl

Author

Roger L. Bagula, Mar 18 2010

Extensions

Edited by G. C. Greubel, Dec 04 2022

Status

approved