A090447 Triangle of partial products of binomials.

1, 1, 1, 1, 2, 2, 1, 3, 9, 9, 1, 4, 24, 96, 96, 1, 5, 50, 500, 2500, 2500, 1, 6, 90, 1800, 27000, 162000, 162000, 1, 7, 147, 5145, 180075, 3781575, 26471025, 26471025, 1, 8, 224, 12544, 878080, 49172480, 1376829440, 11014635520, 11014635520, 1, 9, 324

Offset

0,5

Links

Table of n, a(n) for n=0..47.

W. Lang, First 10 rows.

Formula

a(n, m) = Product_{p=0..m} binomial(n, p), n>=m>=0, else 0. Partial row products in Pascal's triangle A007318.

a(n, m) = (Product_{p=0..m} fallfac(n, m-p))/superfac(m), n>=m>=0, else 0; with fallfac(n, m) := A008279(n, m) (falling factorials) and superfac(m) = A000178(m) (superfactorials).

a(n, m) = (Product_{p=0..m} (n-p)^(m-p))/superfac(m), n>=m>=0, with 0^0:=0, else 0.

Example

[1]; [1,1]; [1,2,2]; [1,3,9,9]; ...

Mathematica

a[n_, m_] := Product[Binomial[n, p], {p, 0, m}]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Sep 01 2016 *)

Crossrefs

Column sequences: A000027 (natural numbers), A006002, A090448-9.

Cf. A090450 (row sums), A090451 (alternating row sums).

Cf. A008949 (partial row sums in Pascal's triangle).

Cf. A000178, A007318, A008279.

Sequence in context: A136203 A113326 A262157 * A241186 A258222 A112324

Adjacent sequences: A090444 A090445 A090446 * A090448 A090449 A090450

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Dec 23 2003

Status

approved