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EXAMPLE
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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).
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PROG
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(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
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