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A090447 Triangle of partial products of binomials. 8
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified May 27 21:14 EDT 2022. Contains 354110 sequences. (Running on oeis4.)