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%I #17 Dec 26 2014 03:03:23
%S 1,4,2,10,12,20,3,42,35,24,112,56,4,108,252,84,40,360,504,120,5,220,
%T 990,924,165,60,880,2376,1584,220,6,390,2860,5148,2574,286,84,1820,
%U 8008,10296,4004,364,7,630,6825,20020,19305,6006,455,112,3360,21840,45760
%N Triangle of numbers a(n,k), n>=3, ceiling((n-3)/2)<=k<=n-3: a(n,k)=Sum[ Binomial[x + y + z, x]*Binomial[y + z, y]*Binomial[n - 2 - x - 2*y - 2*z, 2*n - 2*y - 5 - 2*k]*(2^x)*((-1)^z), {z, 0, (2*k - n + 3)/2}, {y, 0, n - 3 - k}, {x, 0, 2*k - n + 3 - 2*z}].
%C a(n,k) counts the permutations in S_n which have zero occurrences of the pattern 213 and one occurrence of the pattern 123 and k descents.
%D D. Hök, Parvisa mönster i permutationer [Swedish], (2007).
%H Alois P. Heinz, <a href="/A128781/b128781.txt">Rows n = 3..201, flattened</a>
%F a(n,k) = s(n,k)+t(n,k), s(n,k) = a(n-1,k-1), t(n,k) = C(n-2,2*n-5-2*k) + t(n-1,k-1) + s(n-1,k), a(3,0)=t(3,0)=1.
%e Triangle begins:
%e n\k 0 1 2 3 4 5 6
%e ----------------------------------
%e 3 1;
%e 4 . 4;
%e 5 . . 10;
%e 6 . . 12, 20;
%e 7 . . 3, 42, 35;
%e 8 . . . 24, 112, 56;
%e 9 . . . 4, 108, 252, 84;
%Y Diagonal gives A000292.
%K nonn,tabf
%O 3,2
%A David Hoek (david.hok(AT)telia.com), Mar 28 2007
%E Edited by _Peter Bala_, Dec 05 2013
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