A172427 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 = 2, read by rows.

1, 1, 1, 1, -3, 1, 1, 6, 6, 1, 1, -15, 30, -15, 1, 1, 120, 600, 600, 120, 1, 1, -63, 2520, -6300, 2520, -63, 1, 1, 5040, 105840, 2116800, 2116800, 105840, 5040, 1, 1, -255, 428400, -4498200, 35985600, -4498200, 428400, -255, 1, 1, 362880, 30844800, 25909632000, 108820454400, 108820454400, 25909632000, 30844800, 362880, 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 = 2.

Example

Triangle begins as:
  1;
  1,    1;
  1,   -3,      1;
  1,    6,      6,        1;
  1,  -15,     30,      -15,        1;
  1,  120,    600,      600,      120,        1;
  1,  -63,   2520,    -6300,     2520,      -63,      1;
  1, 5040, 105840,  2116800,  2116800,   105840,   5040,    1;
  1, -255, 428400, -4498200, 35985600, -4498200, 428400, -255, 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, 2], {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, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 07 2021

Crossrefs

Cf. this sequence (q=3), A172428 (q=4), A172429 (q=5).

Sequence in context: A157243 A146769 A189610 * A143362 A182823 A210866

Adjacent sequences: A172424 A172425 A172426 * A172428 A172429 A172430

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 02 2010

Extensions

Edited by G. C. Greubel, May 07 2021

Status

approved