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%I #5 Jan 07 2020 17:38:20
%S 1,1,2,7,9,21,76,104,255,936,1321,3327,12250,17682,45200,166420,
%T 244431,630586,2318863,3453662,8964527,32909250,49579091,129250365,
%U 473604092,720390529,1884080667,6890849851,10567693128,27706723888,101151927464,156225285080,410396361463
%N Number of words of length n composed of the letters a, b, and c with at least as many a's as b's, and at least as many b's as c's, and no a's adjacent.
%H Andrew Howroyd, <a href="/A174247/b174247.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = Sum_{i=floor((n+2)/3)..floor((n+1)/2)} Sum_{j=floor((n-i+1)/2)..min(n-i, i)} binomial(n-i+1, i)*binomial(n-i, j). - _Andrew Howroyd_, Jan 07 2020, after Maple
%e For n = 0, there is 1 word (the empty word).
%e For n = 1, there is 1 word: a.
%e For n = 2, there are 2 words: ab, ba.
%e For n = 3, there are 7 words: aba, abc, acb, bac, bca, cab, cba.
%p myseq:=n->add(add(binomial(n-na+1,na)*binomial(n-na,nb),nb=floor((n-na+1)/2)..min(n-na,na)),na=floor((n+2)/3)..floor((n+1)/2));
%o (PARI) a(n)={sum(i=(n+2)\3, (n+1)\2, sum(j=(n-i+1)\2, min(n-i, i), binomial(n-i+1, i)*binomial(n-i, j)))} \\ _Andrew Howroyd_, Jan 07 2020
%K nonn,easy
%O 0,3
%A Amanda Lee (amanda.lee(AT)dpcdsb.org), Mar 13 2010
%E Terms a(11) and beyond from _Andrew Howroyd_, Jan 07 2020
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