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 A123305 Triangle T, read by rows, where column k of T equals (k+1)*(column k of T^2) when shifted to have an initial '1'; i.e., T(n,k) = (k+1)*[T^2](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0. 8
 1, 1, 1, 2, 2, 1, 6, 8, 3, 1, 26, 44, 18, 4, 1, 156, 312, 144, 32, 5, 1, 1234, 2776, 1422, 336, 50, 6, 1, 12340, 30312, 16848, 4256, 650, 72, 7, 1, 150994, 397728, 235458, 63072, 10050, 1116, 98, 8, 1, 2204112, 6151768, 3827628, 1076128, 178900, 20376, 1764 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS FORMULA T(n,k) = (k+1)*Sum_{j=0..n-1} T(n-1,j)*T(j,k) for n>0, with T(n,n)=1 n>=0. EXAMPLE Triangle begins: 1; 1, 1; 2, 2, 1; 6, 8, 3, 1; 26, 44, 18, 4, 1; 156, 312, 144, 32, 5, 1; 1234, 2776, 1422, 336, 50, 6, 1; 12340, 30312, 16848, 4256, 650, 72, 7, 1; 150994, 397728, 235458, 63072, 10050, 1116, 98, 8, 1; ... Matrix square starts: 1; 2, 1; 6, 4, 1; 26, 22, 6, 1; 156, 156, 48, 8, 1; 1234, 1388, 474, 84, 10, 1; 12340, 15156, 5616, 1064, 130, 12, 1; ... Note that (column k of T shifted) = (k+1)*(column k of T^2): k=1: [2,8,44,312,2776,...] = 2*[1,4,22,156,1388,...]; k=2: [3,18,144,1422,16848,....] = 3*[1,6,48,474,5616,...]. PROG (PARI) {T(n, k)=if(n<0|k>n, 0, if(n==k, 1, (k+1)*sum(j=0, n-1, T(n-1, j)*T(j, k)); ))} CROSSREFS Columns: A123306, A123307, A123308, A123309; A123310 (row sums); A123311 (central terms). Sequence in context: A222073 A135880 A077873 * A118024 A184184 A074297 Adjacent sequences:  A123302 A123303 A123304 * A123306 A123307 A123308 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Sep 24 2006 STATUS approved

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