A073617 Consider Pascal's triangle A007318; a(n) = product of terms at +45 degrees slope with the horizontal.

1, 1, 1, 2, 3, 12, 30, 240, 1050, 16800, 132300, 4233600, 61122600, 3911846400, 104886381600, 13425456844800, 674943865596000, 172785629592576000, 16407885372638760000, 8400837310791045120000, 1515727634953623371280000, 1552105098192510332190720000

Offset

0,4

Comments

The sum of the terms pertaining to the above product is the (n+1)-th Fibonacci number: 1 + 5 + 6 + 1 = 13.

n divides A073617(n+1) for n>=1; see the Mathematica section. [Clark Kimberling, Feb 29 2012]

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = Product_{k=0..floor(n/2)} binomial(n-k,k).

a(2n+1)/a(2n-1) = binomial(2n,n); a(2n)/a(2n-2) = (1/2)*binomial(2n,n); (a(2n+1)*a(2n-2))/(a(2n)*a(2n-1))] = 2. - John Molokach, Sep 09 2013

Example

For n=6 the diagonal is 1,5,6,1 and product of the terms = 30 hence a(6) = 30.

Maple

a:= n-> mul(binomial(n-i, i), i=0..floor(n/2)):
seq(a(n), n=0..21);  # Alois P. Heinz, Nov 27 2023

Mathematica

p[n_] := Product[Binomial[n + 1 - k, k], {k, 1, Floor[(n + 1)/2]}]
Table[p[n], {n, 1, 20}]   (* A073617(n+1) *)
Table[p[n]/n, {n, 1, 20}] (* A208649 *)
( * Clark Kimberling, Feb 29 2012 *)

Crossrefs

Cf. A000045, A073618, A007685, A208649, A000984.

Sequence in context: A105401 A325595 A256031 * A034381 A076424 A165301

Adjacent sequences: A073614 A073615 A073616 * A073618 A073619 A073620

Keyword

nonn

Author

Amarnath Murthy, Aug 07 2002

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 22 2003

Status

approved