A103410 Number of products of distinct elements in generation n, starting with two elements.

2, 1, 2, 7, 56, 2212, 2595782, 3374959180831, 5695183504489239067484387, 16217557574922386301420531277071365103168734284282, 131504586847961235687181874578063117114329409897598970946516793776220805297959867258692249572750581

Offset

0,1

Comments

The binary operation must be commutative, idempotent and non-associative. - David Wasserman, Apr 15 2008

Links

Table of n, a(n) for n=0..10.

Formula

a(n)=a(n-1)(a(0)+a(1)+...+a(n-2))+C(a(n-1), 2).

Example

The word "product" means a binary operation * . For example, using * = average, given by a*b=(a+b)/2, generation G(0) consisting of 0 and 1 yields successive generations:
G(1): 0*1=1/2, whence a(1)=1
G(2): 1/4=0*(1/2), 3/4=1*(1/2), whence a(2)=2
G(3): 1/8=0*(1/4), 5/8=1*(1/4), 3/8=(1/2)*(1/4), 3/8=0*(3/4),
7/8=1*(3/4), 5/8=(1/2)*(3/4), 1/2=(1/4)*(3/4), whence a(3)=7.
To summarize, for n>=3, G(n) consists of a(n-1)*(a(0)+a(1)+...+a(n-2)) products a*b where a runs through G(0), G(1),...,G(n-2) and b runs through G(n-1), together with C(a(n-1),2) products a*b where a and b run through G(n-1).

Prog

(PARI) print1("2, "); a=2; s=0; for(n=1, 12, aa=a*s+binomial(a, 2); print1(aa", "); s+=a; a=aa) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

Crossrefs

The same as A002658 for n >= 1.

Sequence in context: A144803 A095062 A032068 * A114303 A030651 A179946

Adjacent sequences: A103407 A103408 A103409 * A103411 A103412 A103413

Keyword

nonn

Author

Clark Kimberling, Feb 04 2005

Extensions

One more term from David Wasserman, Apr 15 2008

One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

Status

approved