A173505 -id:A173505 - cp's OEIS frontend

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

Roger L. Bagula, Feb 20 2010

			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;

G. C. Greubel, Rows n = 0..29 of the triangle, flattened

Cf. A173504, A173505.

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 *)

@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

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.

Edited by G. C. Greubel, Apr 25 2021

A173504 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 = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 16, 16, 1, 1, 416, 3328, 416, 1, 1, 33280, 6922240, 6922240, 33280, 1, 1, 8053760, 134014566400, 3484378726400, 134014566400, 8053760, 1, 1, 5863137280, 23610150250086400, 49109112520179712000, 49109112520179712000, 23610150250086400, 5863137280, 1

Offset: 0

Roger L. Bagula, Feb 20 2010

			The triangle begins as:
  1;
  1,       1;
  1,       2,            1;
  1,      16,           16,             1;
  1,     416,         3328,           416,            1;
  1,   33280,      6922240,       6922240,        33280,       1;
  1, 8053760, 134014566400, 3484378726400, 134014566400, 8053760, 1;

G. C. Greubel, Rows n = 0..23 of the triangle, flattened

Cf. A173503, A173505.

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, 3], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 25 2021 *)

@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,3) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Apr 25 2021

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 = 3.

Edited by G. C. Greubel, Apr 25 2021

cp's OEIS Frontend

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.

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