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A228708 Triangle T(n,k) read by rows: T(n,k) = number of permutations on 123...n with exactly one abc pattern and no aj pattern with j<=k, for n>=0, 0<=k<=n.

%I A228708 #13 Sep 15 2013 13:24:17
%S A228708 0,0,0,0,0,0,1,1,0,0,6,6,2,0,0,27,27,12,3,0,0,110,110,55,19,4,0,0,429,
%T A228708 429,229,91,27,5,0,0,1638,1638,912,393,136,36,6,0,0,6188,6188,3549,
%U A228708 1614,612,191,46,7,0,0,23256,23256,13636,6447,2601,897,257,57,8,0,0
%N A228708 Triangle T(n,k) read by rows: T(n,k) = number of permutations on 123...n with exactly one abc pattern and no aj pattern with j<=k, for n>=0, 0<=k<=n.
%C A228708 See Noonan-Zeilberger for precise definition.
%H A228708 J. Noonan and D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/9808080">[math/9808080] The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns</a>. Also Adv. in Appl. Math. 17 (1996), no. 4, 381--407. MR1422065 (97j:05003).
%F A228708 T(n, k) = C(2n-k-1, n) - C(2n-k-1, n+3) + C(2n-2k-2, n-k-4) - C(2n-2k-2, n-k-1) + C(2n-2k-3, n-k-4) - C(2n-2k-3, n-k-2).
%F A228708 T(n, n-2) = n-2, T(n, k) = T(n, k+1) + T(n-1, k-1) + T(n-k, 2).
%e A228708 Triangle begins:
%e A228708 0
%e A228708 0,0
%e A228708 0,0,0
%e A228708 1,1,0,0
%e A228708 6,6,2,0,0
%e A228708 27,27,12,3,0,0
%e A228708 110,110,55,19,4,0,0
%e A228708 429,429,229,91,27,5,0,0
%e A228708 1638,1638,912,393,136,36,6,0,0
%e A228708 6188,6188,3549,1614,612,191,46,7,0,0
%e A228708 23256,23256,13636,6447,2601,897,257,57,8,0,0
%e A228708 ...
%o A228708 (PARI) for(n=1,15, for(k=1,n-2,print1(binomial(2*n-k-1,n)-binomial(2*n-k-1,n+3)+binomial(2*n-2*k-2,n-k-4)-binomial(2*n-2*k-2,n-k-1)+binomial(2*n-2*k-3,n-k-4)-binomial(2*n-2*k-3,n-k-2)",")))
%Y A228708 See A084249 for a curtailed version. See also A229158, A229160.
%Y A228708 T(n, 1) = A003517(n+1). Cf. A001089.
%K A228708 nonn,tabl,easy
%O A228708 0,11
%A A228708 _N. J. A. Sloane_, Sep 15 2013