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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
The same as A002658 for n >= 1.
Sequence in context: A144803 A095062 A032068 * A114303 A030651 A179946
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 3 05:18 EDT 2024. Contains 375649 sequences. (Running on oeis4.)