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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. 4
1, -1, 1, 1, -2, 1, -2, 4, -3, 1, 5, -11, 9, -4, 1, -17, 38, -33, 16, -5, 1, 71, -162, 145, -74, 25, -6, 1, -357, 824, -753, 396, -140, 36, -7, 1, 2101, -4892, 4535, -2434, 885, -237, 49, -8, 1, -14203, 33286, -31185, 16982, -6295, 1730, -371, 64, -9, 1, 108609, -255824, 241621, -133012, 50001, -13992, 3073, -548, 81 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Column 0 forms A101900. Absolute row sums form A101901.
LINKS
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.
FORMULA
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.
EXAMPLE
Rows begin:
1;
-1, 1;
1, -2, 1;
-2, 4, -3, 1;
5, -11, 9, -4, 1;
-17, 38, -33, 16, -5, 1;
71, -162, 145, -74, 25, -6, 1;
-357, 824, -753, 396, -140, 36, -7, 1,
2101, -4892, 4535, -2434, 885, -237, 49, -8, 1;
-14203, 33286, -31185, 16982, -6295, 1730, -371, 64, -9, 1;
...
MATHEMATICA
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 *)
PROG
(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)); )); )}
CROSSREFS
Sequence in context: A339549 A179750 A091173 * A208058 A078142 A133422
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Dec 20 2004
STATUS
approved

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Last modified April 12 17:42 EDT 2024. Contains 371635 sequences. (Running on oeis4.)