A090447 - OEIS

%I #14 Aug 28 2019 16:55:27

%S 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,

%T 27000,162000,162000,1,7,147,5145,180075,3781575,26471025,26471025,1,

%U 8,224,12544,878080,49172480,1376829440,11014635520,11014635520,1,9,324

%N Triangle of partial products of binomials.

%H W. Lang, <a href="/A090447/a090447.txt">First 10 rows</a>.

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

%F 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).

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

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

%t 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 *)

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

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

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

%Y Cf. A000178, A007318, A008279.

%K nonn,easy,tabl

%O 0,5

%A _Wolfdieter Lang_, Dec 23 2003