A109714 Sequence defined by a recurrence close to that of A001147.

1, 1, 3, 18, 120, 1170, 12600, 176400, 2608200, 46607400, 883159200, 19429502400, 447567120000, 11629447830000, 316028116404000, 9516436753824000, 297478346845680000, 10151626256147376000, 359237701318479984000, 13733349319337487840000, 542212802070902202240000

Offset: 1

Author

Niko Brummer (niko.brummer(AT)gmail.com), Aug 08 2005

Keywords

• easy
• nonn

Comments

From Christopher J. Smyth, Jan 26 2018: (Start)

The sequence is defined by the recurrence formula below. This recurrence is very similar to that of the sequence b(n) = A001147(n-1), which satisfies b(1)=1 and, for n >= 2, b(n) = Sum_{i=1..floor((n-1)/2)} binomial(n, i) * b(i) * b(n-i) + B, where B = 0 (n odd), = (1/2)*binomial(n, n/2)*b(n/2)^2 (n even) [see formula of Walsh on A001147 page]. Removal of the factor 1/2 from the definition of B gives, for n >= 3, the formula below for a(n).

This sequence seems to have been defined in the mistaken belief that it had applications. In fact the applications stated on earlier versions of this page actually belonged to A001147 -- see my comment on the A001147 page.

(End)

Examples

			a(3) = 3*a(1)*a(2) = 3, a(4) = 4*a(1)*a(3) + 6*a(2)^2 = 18.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..403

Crossrefs

Cf. A001147

Programs

MATLAB

function m = a(n); if n==1 m = 1; elseif n==2 m = 1; else m = 0; for i=1:floor(n/2); f1 = binomial(n,i); f2 = a(i); f3 = a(n-i); m = m + f1*f2*f3; end; end;

Mathematica

Fold[Append[#1, Sum[Binomial[#2, i] #1[[i]] #1[[#2 - i]], {i, Floor[#2/2]}]] &, {1, 1}, Range[3, 21]] (* Michael De Vlieger, Dec 13 2017 *)

Formula

a(1) = 1, a(2) = 1 and a(n) = Sum_{i=1..floor(n/2)} binomial(n, i) * a(i) * a(n-i) for n >= 3.