A173503 Triangle T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) where c(n,q) = Product_{j=1..n} (q^j -1)^(n-j) and q = 2, read by rows.

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 21, 63, 21, 1, 1, 315, 6615, 6615, 315, 1, 1, 9765, 3075975, 21531825, 3075975, 9765, 1, 1, 615195, 6007379175, 630774813375, 630774813375, 6007379175, 615195, 1, 1, 78129765, 48065040779175, 156451707736214625, 2346775616043219375, 156451707736214625, 48065040779175, 78129765, 1

Offset

0,8

Links

G. C. Greubel, Rows n = 0..29 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} (q^j -1)^(n-j) and q = 2.

Example

The triangle begins as:
  1;
  1,      1;
  1,      1,          1;
  1,      3,          3,            1;
  1,     21,         63,           21,            1;
  1,    315,       6615,         6615,          315,          1;
  1,   9765,    3075975,     21531825,      3075975,       9765,      1;
  1, 615195, 6007379175, 630774813375, 630774813375, 6007379175, 615195, 1;

Mathematica

c[n_, q_]:= Product[(q^m-1)^(n-m), {m, 1, n}];
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n, k, 2], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 25 2021 *)

Prog

(Sage)
@CachedFunction
def c(n, q): return product( (q^j -1)^(n-j) 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..10)]) # G. C. Greubel, Apr 25 2021

Crossrefs

Cf. A173504, A173505.

Sequence in context: A087107 A155170 A126460 * A338114 A100940 A344390

Adjacent sequences: A173500 A173501 A173502 * A173504 A173505 A173506

Keyword

nonn,tabl,less

Author

Roger L. Bagula, Feb 20 2010

Extensions

Edited by G. C. Greubel, Apr 25 2021

Status

approved