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A131085 Triangle T(n,k) (n>=0, 0<=k<=n-1) read by rows, A007318 * A129686.

%I A131085 #35 Feb 11 2019 21:13:44
%S A131085 1,1,1,0,2,1,-2,2,3,1,-5,0,5,4,1,-9,-5,5,9,5,1,-14,-14,0,14,14,6,1,
%T A131085 -20,-28,-14,14,28,20,7,1,-27,-48,-42,0,42,48,27,8,1,-35,-75,-90,-42,
%U A131085 42,90,75,35,9,1,-44,-110,-165,-132,0,132,165,110,44,10,1
%N A131085 Triangle T(n,k) (n>=0, 0<=k<=n-1) read by rows, A007318 * A129686.
%C A131085 Row sums = n.
%C A131085 A131085 * A000012 = A074909 starting (1, 2, 1, 3, 3, ...) instead of (1, 1, 2, 1, 3, 3, ...).
%F A131085 Binomial transform of A129686 signed with (1, 1, 1, ...) in the main diagonal and (-1, -1, -1, ...) in the subsubdiagonal.
%F A131085 T(n,m) = T(n-1,m-1) + T(n-1,m). - _Yuchun Ji_, Dec 17 2018
%F A131085 T(2*k,k-1) = 0 for k > 0. - _Yuchun Ji_, Dec 20 2018
%F A131085 Comparing this triangle with the Catalan triangle A009766 one can see many similarities. For example, T(k+j,k) = A009766(k+1,j) for j < k+2. - _Yuchun Ji_, Dec 23 2018 [Edited by _N. J. A. Sloane_, Feb 11 2019]
%e A131085 First few rows of the triangle are:
%e A131085    1;
%e A131085    1,  1;
%e A131085    0,  2, 1;
%e A131085   -2,  2, 3, 1;
%e A131085   -5,  0, 5, 4, 1;
%e A131085   -9, -5, 5, 9, 5, 1;
%e A131085 -14, -14, 0, 14, 14, 6, 1;
%e A131085 -20, -28, -14, 14, 28, 20, 7, 1;
%e A131085 -27, -48, -42, 0, 42, 48, 27, 8, 1;
%e A131085 -35, -75, -90, -42, 42, 90, 75, 35, 9, 1;
%e A131085    ...
%o A131085 (PARI) tabl(nn) = {t007318 = matrix(nn, nn, n, k, binomial(n-1, k-1)); t129686 = matrix(nn, nn, n, k, (k<=n)*((-1)^((n-k)\2)*((k==n) || (-1)*(k==(n-2))))); t131085 = t007318*t129686; for (n = 1, nn, for (k = 1, n, print1(t131085[n, k], ", ");););} \\ _Michel Marcus_, Feb 12 2014
%Y A131085 Cf. A007318, A009766, A097808, A112466, A214292, A112467, A129686, A131084.
%K A131085 sign,tabl
%O A131085 0,5
%A A131085 _Gary W. Adamson_, Jun 14 2007
%E A131085 Missing comma corrected by Naruto Canada, Aug 26 2007
%E A131085 More terms from _Michel Marcus_, Feb 12 2014
%E A131085 Offset changed by _N. J. A. Sloane_, Feb 11 2019