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A101897 Triangle T, read by rows, such that column k equals column 0 of T^(k+1), where column 0 of T allows the n-th row sums to be zero for n>0 and where T^k is the k-th power of T as a lower triangular matrix.

%I A101897 #19 Jul 19 2024 19:04:35
%S A101897 1,-1,1,1,-2,1,-2,4,-3,1,5,-11,9,-4,1,-17,38,-33,16,-5,1,71,-162,145,
%T A101897 -74,25,-6,1,-357,824,-753,396,-140,36,-7,1,2101,-4892,4535,-2434,885,
%U A101897 -237,49,-8,1,-14203,33286,-31185,16982,-6295,1730,-371,64,-9,1,108609,-255824,241621,-133012,50001,-13992,3073,-548,81
%N A101897 Triangle T, read by rows, such that column k equals column 0 of T^(k+1), where column 0 of T allows the n-th row sums to be zero for n>0 and where T^k is the k-th power of T as a lower triangular matrix.
%C A101897 Column 0 forms A101900. Absolute row sums form A101901.
%H A101897 J.-L. Baril, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p178">Classical sequences revisited with permutations avoiding dotted pattern</a>, Electronic Journal of Combinatorics, 18 (2011), #P178.
%F A101897 T(n, k) = Sum_{j=0..n-k} T(n-k, j)*T(j+k-1, k-1) for n >= k > 0 with T(0, 0) = 1 and T(n, 0) = -Sum_{j=1, n} T(n, j) for n > 0.
%e A101897 Rows begin:
%e A101897         1;
%e A101897        -1,     1;
%e A101897         1,    -2,      1;
%e A101897        -2,     4,     -3,     1;
%e A101897         5,   -11,      9,    -4,     1;
%e A101897       -17,    38,    -33,    16,    -5,    1;
%e A101897        71,  -162,    145,   -74,    25,   -6,   1;
%e A101897      -357,   824,   -753,   396,  -140,   36,  -7,   1,
%e A101897      2101, -4892,   4535, -2434,   885, -237,  49,  -8,  1;
%e A101897    -14203, 33286, -31185, 16982, -6295, 1730, -371, 64, -9, 1;
%e A101897       ...
%t A101897 t[n_, k_] := t[n, k] = If[k>n || n<0 || k<0, 0, If[k==n, 1, If[k==0, -Sum[t[n, j], {j, 1, n}], Sum[t[n-k, j]*t[j+k-1, k-1], {j, 0, n-k}]]]]; Table[t[n ,k], {n,0,10}, {k, 0, n}] //Flatten (* _Amiram Eldar_, Nov 26 2018 *)
%o A101897 (PARI) {T(n,k)=if(k>n||n<0||k<0,0,if(k==n,1, if(k==0,-sum(j=1,n,T(n,j)), sum(j=0,n-k,T(n-k,j)*T(j+k-1,k-1));));)}
%Y A101897 Cf. A091351, A101900, A101901, A101898, A101899.
%K A101897 sign,tabl
%O A101897 0,5
%A A101897 _Paul D. Hanna_, Dec 20 2004