A172429 Triangle T(n, k, q) = c(n,q)/( c(k,q)*c(n-k,q) ), where c(n, q) = Product_{j=1..n} f(n, q), f(n, q) = ( (1-q^n)*(1+(-1)^n) + n!*(1-(-1)^n) )/2, and q = 4, read by rows.

1, 1, 1, 1, -15, 1, 1, 6, 6, 1, 1, -255, 102, -255, 1, 1, 120, 2040, 2040, 120, 1, 1, -4095, 32760, -1392300, 32760, -4095, 1, 1, 5040, 1375920, 27518400, 27518400, 1375920, 5040, 1, 1, -65535, 22019760, -15028486200, 7072228800, -15028486200, 22019760, -65535, 1

Offset

0,5

Links

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

Formula

T(n, k, q) = c(n,q)/( c(k,q)*c(n-k,q) ), where c(n, q) = Product_{j=1..n} f(n, q), f(n, q) = ( (1-q^n)*(1+(-1)^n) + n!*(1-(-1)^n) )/2, and q = 4.

Example

The triangle begins as:
  1;
  1,      1;
  1,    -15,        1;
  1,      6,        6,            1;
  1,   -255,      102,         -255,          1;
  1,    120,     2040,         2040,        120,            1;
  1,  -4095,    32760,     -1392300,      32760,        -4095,        1;
  1,   5040,  1375920,     27518400,   27518400,      1375920,     5040,      1;
  1, -65535, 22019760, -15028486200, 7072228800, -15028486200, 22019760, -65535, 1;

Mathematica

f[n_, q_]:= ((1-q^n)*(1+(-1)^n) + n!*(1-(-1)^n))/2;
c[n_, q_]:= Product[f[j, q], {j, n}];
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n, k, 4], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 07 2021 *)

Prog

(Sage)
@CachedFunction
def f(n, q): return ((1-q^n)*(1+(-1)^n) + factorial(n)*(1-(-1)^n))/2
def c(n, q): return product( f(j, q) for j in (1..n) )
def T(n, k, q): return c(n, q)/(c(k, q)*c(n-k, q))
flatten([[T(n, k, 4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 07 2021

Crossrefs

Cf. A172427 (q=3), A172428 (q=4), this sequence (q=5).

Sequence in context: A281571 A040227 A040226 * A040225 A070644 A174389

Adjacent sequences: A172426 A172427 A172428 * A172430 A172431 A172432

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 02 2010

Extensions

Edited by G. C. Greubel, May 07 2021

Status

approved