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 A152400 Triangle T, read by rows, where column k of T = column 0 of matrix power T^(k+1) for k>0, with column 0 of T = unsigned column 0 of T^-1 (shifted). 6
 1, 1, 1, 3, 2, 1, 14, 8, 3, 1, 86, 45, 15, 4, 1, 645, 318, 99, 24, 5, 1, 5662, 2671, 794, 182, 35, 6, 1, 56632, 25805, 7414, 1636, 300, 48, 7, 1, 633545, 280609, 78507, 16844, 2990, 459, 63, 8, 1, 7820115, 3381993, 926026, 194384, 33685, 5026, 665, 80, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Table of n, a(n) for n=0..54. FORMULA Column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0. Column k: T(n,k) = Sum_{j=0..n-k} T(n-k,j)*T(j+k-1,k-1) for n>=k>0. Column 0: T(n,0) = Sum_{j=1..n} T(n,j)*T(j-1,0) for n>=0. EXAMPLE Triangle T begins: 1; 1, 1; 3, 2, 1; 14, 8, 3, 1; 86, 45, 15, 4, 1; 645, 318, 99, 24, 5, 1; 5662, 2671, 794, 182, 35, 6, 1; 56632, 25805, 7414, 1636, 300, 48, 7, 1; 633545, 280609, 78507, 16844, 2990, 459, 63, 8, 1; 7820115, 3381993, 926026, 194384, 33685, 5026, 665, 80, 9, 1;... where column k of T = column 0 of T^(k+1) for k>0 and column 0 of T = unsigned column 0 of T^-1 (shifted). Amazingly, column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0. Matrix inverse T^-1 begins: 1; -1, 1; -1, -2, 1; -3, -2, -3, 1; -14, -7, -3, -4, 1; -86, -37, -12, -4, -5, 1; -645, -252, -71, -18, -5, -6, 1;... where unsigned column 0 of T^-1 = column 0 of T (shifted). Matrix square T^2 begins: 1; 2, 1; 8, 4, 1; 45, 22, 6, 1; 318, 152, 42, 8, 1; 2671, 1251, 345, 68, 10, 1; 25805, 11869, 3253, 648, 100, 12, 1; 280609, 126987, 34546, 6898, 1085, 138, 14, 1;... where column 0 of T^2 = column 1 of T, and column 2 of T^2 = column 1 of T^3. Matrix cube T^3 begins: 1; 3, 1; 15, 6, 1; 99, 42, 9, 1; 794, 345, 81, 12, 1; 7414, 3253, 798, 132, 15, 1; 78507, 34546, 8679, 1518, 195, 18, 1; 926026, 407171, 103707, 18734, 2565, 270, 21, 1;... where column 0 of T^3 = column 2 of T, and column 3 of T^3 = column 2 of T^4. Matrix power T^4 begins: 1; 4, 1; 24, 8, 1; 182, 68, 12, 1; 1636, 648, 132, 16, 1; 16844, 6898, 1518, 216, 20, 1; 194384, 81218, 18734, 2912, 320, 24, 1; 2476868, 1047638, 249202, 40932, 4950, 444, 28, 1;... where column 0 of T^4 = column 3 of T, and column 2 of T^4 = column 3 of T^3. Related triangle A127714 begins: 1; 1, 1, 1; 1, 2, 2, 3, 3, 3; 1, 3, 5, 5, 8, 11, 11, 14, 14, 14; 1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86;... where right border = column 0 of this triangle A152400. PROG (PARI) T(n, k)=if(k>n || n<0, 0, if(k==n, 1, if(k==0, sum(j=1, n, T(n, j)*T(j-1, 0)), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); )) CROSSREFS Cf. columns: A127715, A152401, A152402, A152403, A152404. Cf. related triangles: A152405, A127714. Sequence in context: A161133 A112911 A152405 * A291978 A342217 A111548 Adjacent sequences: A152397 A152398 A152399 * A152401 A152402 A152403 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Dec 05 2008 STATUS approved

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Last modified September 25 13:12 EDT 2023. Contains 365647 sequences. (Running on oeis4.)