This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A103410 #12 Oct 05 2024 19:15:49
%S A103410 2,1,2,7,56,2212,2595782,3374959180831,5695183504489239067484387,
%T A103410 16217557574922386301420531277071365103168734284282,
%U A103410 131504586847961235687181874578063117114329409897598970946516793776220805297959867258692249572750581
%N A103410 Number of products of distinct elements in generation n, starting with two elements.
%C A103410 The binary operation must be commutative, idempotent and non-associative. - _David Wasserman_, Apr 15 2008
%F A103410 a(n)=a(n-1)(a(0)+a(1)+...+a(n-2))+C(a(n-1), 2).
%e A103410 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:
%e A103410 G(1): 0*1=1/2, whence a(1)=1
%e A103410 G(2): 1/4=0*(1/2), 3/4=1*(1/2), whence a(2)=2
%e A103410 G(3): 1/8=0*(1/4), 5/8=1*(1/4), 3/8=(1/2)*(1/4), 3/8=0*(3/4),
%e A103410 7/8=1*(3/4), 5/8=(1/2)*(3/4), 1/2=(1/4)*(3/4), whence a(3)=7.
%e A103410 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).
%o A103410 (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
%Y A103410 The same as A002658 for n >= 1.
%K A103410 nonn
%O A103410 0,1
%A A103410 _Clark Kimberling_, Feb 04 2005
%E A103410 One more term from _David Wasserman_, Apr 15 2008
%E A103410 One more term from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008