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 A103410 Number of products of distinct elements in generation n, starting with two elements. 1
 2, 1, 2, 7, 56, 2212, 2595782, 3374959180831, 5695183504489239067484387, 16217557574922386301420531277071365103168734284282, 131504586847961235687181874578063117114329409897598970946516793776220805297959867258692249572750581 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The binary operation must be commutative, idempotent and non-associative. - David Wasserman, Apr 15 2008 LINKS 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 Essentially the same as A002658. 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

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Last modified September 18 04:22 EDT 2021. Contains 347508 sequences. (Running on oeis4.)