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 A152391 Triangle T, read by rows, where the matrix square T^2 results in shifting T right one column to drop the secondary diagonal. 8
 1, 1, 1, 4, 2, 1, 18, 6, 3, 1, 96, 28, 8, 4, 1, 580, 150, 40, 10, 5, 1, 3852, 930, 216, 54, 12, 6, 1, 27678, 6286, 1386, 294, 70, 14, 7, 1, 212224, 46120, 9552, 1960, 384, 88, 16, 8, 1, 1722312, 359946, 71820, 13770, 2664, 486, 108, 18, 9, 1, 14685140, 2973650, 571440, 106290, 19060, 3510, 600, 130, 20, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..350 FORMULA T(n, k) = Sum_{j=k+1..n} T(n,j) * T(j,k+1) for n > k+1 >= 1 with T(n+1,n)=n+1 and T(n,n)=1 for n >= 0. EXAMPLE Triangle T begins: 1; 1, 1; 4, 2, 1; 18, 6, 3, 1; 96, 28, 8, 4, 1; 580, 150, 40, 10, 5, 1; 3852, 930, 216, 54, 12, 6, 1; 27678, 6286, 1386, 294, 70, 14, 7, 1; 212224, 46120, 9552, 1960, 384, 88, 16, 8, 1; 1722312, 359946, 71820, 13770, 2664, 486, 108, 18, 9, 1; 14685140, 2973650, 571440, 106290, 19060, 3510, 600, 130, 20, 10, 1; ... Illustrate recurrence by products of row and column vectors: T(4,1) = [8,4,1]*[1,3,8]~ = 8*1 + 4*3 + 1*8 = 28; T(6,0) = [930,216,54,12,6,1]*[1,2,6,28,150,930]~ = 3852; T(7,0) = [6286,1386,294,70,14,7,1]*[1,2,6,28,150,930,6286]~ = 27678. T(8,1) = [9552,1960,384,88,16,8,1]*[1,3,8,40,216,1386,9552]~ = 46120. T(9,3) = [2664,486,108,18,9,1]*[1,5,12,70,384,2664]~ = 13770. Matrix square T^2 begins: 1; 2, 1; 10, 4, 1; 54, 18, 6, 1; 324, 96, 28, 8, 1; 2130, 580, 150, 40, 10, 1; 15102, 3852, 930, 216, 54, 12, 1; 114282, 27678, 6286, 1386, 294, 70, 14, 1; ... which equals T shifted right one column with the secondary diagonal dropped. PROG (PARI) {T(n, k) = if(n==k, 1, if(n==k+1, n, sum(j=k+1, n, T(n, j)*T(j, k+1) )))} for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) (PARI) /* Build an N X N Matrix (informal) */ {M = matrix(N, N, n, k, if(n==k, 1, if(n==k+1, n)) ); } {T(n, k) = M[n+1, k+1] = if(n==k, 1, if(n==k+1, n, sum(j=k+1, n, T(n, j) * M[j+1, k+2] )))} for(n=0, N, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Jan 13 2016 CROSSREFS Cf. columns: A152392, A152393, A152394; A152395. Cf. A109152. Sequence in context: A269736 A264535 A256039 * A144088 A039948 A111536 Adjacent sequences: A152388 A152389 A152390 * A152392 A152393 A152394 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Dec 11 2008 STATUS approved

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Last modified December 8 06:27 EST 2023. Contains 367662 sequences. (Running on oeis4.)