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A084249 Triangle T(n,k) read by rows: permutations on 123...n with one abc pattern and no aj pattern with j<=k, n>2, k

%I A084249 #9 Sep 15 2013 12:44:00
%S A084249 1,6,2,27,12,3,110,55,19,4,429,229,91,27,5,1638,912,393,136,36,6,6188,
%T A084249 3549,1614,612,191,46,7,23256,13636,6447,2601,897,257,57,8,87210,
%U A084249 52020,25332,10695,3951,1260,335,69,9,326876,197676,98532
%N A084249 Triangle T(n,k) read by rows: permutations on 123...n with one abc pattern and no aj pattern with j<=k, n>2, k<n-1.
%C A084249 See A228708 for further information.
%H A084249 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 A084249 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 A084249 T(n, n-2) = n-2, T(n, k) = T(n, k+1) + T(n-1, k-1) + T(n-k, 2).
%e A084249 Full triangle begins:
%e A084249 0
%e A084249 0,0
%e A084249 0,0,0
%e A084249 1,1,0,0
%e A084249 6,6,2,0,0
%e A084249 27,27,12,3,0,0
%e A084249 110,110,55,19,4,0,0
%e A084249 429,429,229,91,27,5,0,0
%e A084249 1638,1638,912,393,136,36,6,0,0
%e A084249 6188,6188,3549,1614,612,191,46,7,0,0
%e A084249 23256,23256,13636,6447,2601,897,257,57,8,0,0
%e A084249 ...
%o A084249 (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 A084249 See A228708 for the full triangle.
%Y A084249 T(n, 1) = A003517(n+1). Cf. A001089.
%K A084249 nonn,tabl,easy
%O A084249 3,2
%A A084249 _Ralf Stephan_, May 21 2003