cp's OEIS Frontend

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).

%I A152400 #4 Jun 14 2017 01:11:09
%S A152400 1,1,1,3,2,1,14,8,3,1,86,45,15,4,1,645,318,99,24,5,1,5662,2671,794,
%T A152400 182,35,6,1,56632,25805,7414,1636,300,48,7,1,633545,280609,78507,
%U A152400 16844,2990,459,63,8,1,7820115,3381993,926026,194384,33685,5026,665,80,9,1
%N 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).
%F A152400 Column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.
%F A152400 Column k: T(n,k) = Sum_{j=0..n-k} T(n-k,j)*T(j+k-1,k-1) for n>=k>0.
%F A152400 Column 0: T(n,0) = Sum_{j=1..n} T(n,j)*T(j-1,0) for n>=0.
%e A152400 Triangle T begins:
%e A152400 1;
%e A152400 1, 1;
%e A152400 3, 2, 1;
%e A152400 14, 8, 3, 1;
%e A152400 86, 45, 15, 4, 1;
%e A152400 645, 318, 99, 24, 5, 1;
%e A152400 5662, 2671, 794, 182, 35, 6, 1;
%e A152400 56632, 25805, 7414, 1636, 300, 48, 7, 1;
%e A152400 633545, 280609, 78507, 16844, 2990, 459, 63, 8, 1;
%e A152400 7820115, 3381993, 926026, 194384, 33685, 5026, 665, 80, 9, 1;...
%e A152400 where column k of T = column 0 of T^(k+1) for k>0
%e A152400 and column 0 of T = unsigned column 0 of T^-1 (shifted).
%e A152400 Amazingly, column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.
%e A152400 Matrix inverse T^-1 begins:
%e A152400 1;
%e A152400 -1, 1;
%e A152400 -1, -2, 1;
%e A152400 -3, -2, -3, 1;
%e A152400 -14, -7, -3, -4, 1;
%e A152400 -86, -37, -12, -4, -5, 1;
%e A152400 -645, -252, -71, -18, -5, -6, 1;...
%e A152400 where unsigned column 0 of T^-1 = column 0 of T (shifted).
%e A152400 Matrix square T^2 begins:
%e A152400 1;
%e A152400 2, 1;
%e A152400 8, 4, 1;
%e A152400 45, 22, 6, 1;
%e A152400 318, 152, 42, 8, 1;
%e A152400 2671, 1251, 345, 68, 10, 1;
%e A152400 25805, 11869, 3253, 648, 100, 12, 1;
%e A152400 280609, 126987, 34546, 6898, 1085, 138, 14, 1;...
%e A152400 where column 0 of T^2 = column 1 of T,
%e A152400 and column 2 of T^2 = column 1 of T^3.
%e A152400 Matrix cube T^3 begins:
%e A152400 1;
%e A152400 3, 1;
%e A152400 15, 6, 1;
%e A152400 99, 42, 9, 1;
%e A152400 794, 345, 81, 12, 1;
%e A152400 7414, 3253, 798, 132, 15, 1;
%e A152400 78507, 34546, 8679, 1518, 195, 18, 1;
%e A152400 926026, 407171, 103707, 18734, 2565, 270, 21, 1;...
%e A152400 where column 0 of T^3 = column 2 of T,
%e A152400 and column 3 of T^3 = column 2 of T^4.
%e A152400 Matrix power T^4 begins:
%e A152400 1;
%e A152400 4, 1;
%e A152400 24, 8, 1;
%e A152400 182, 68, 12, 1;
%e A152400 1636, 648, 132, 16, 1;
%e A152400 16844, 6898, 1518, 216, 20, 1;
%e A152400 194384, 81218, 18734, 2912, 320, 24, 1;
%e A152400 2476868, 1047638, 249202, 40932, 4950, 444, 28, 1;...
%e A152400 where column 0 of T^4 = column 3 of T,
%e A152400 and column 2 of T^4 = column 3 of T^3.
%e A152400 Related triangle A127714 begins:
%e A152400 1;
%e A152400 1, 1, 1;
%e A152400 1, 2, 2, 3, 3, 3;
%e A152400 1, 3, 5, 5, 8, 11, 11, 14, 14, 14;
%e A152400 1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86;...
%e A152400 where right border = column 0 of this triangle A152400.
%o A152400 (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)));))
%Y A152400 Cf. columns: A127715, A152401, A152402, A152403, A152404.
%Y A152400 Cf. related triangles: A152405, A127714.
%K A152400 nonn,tabl
%O A152400 0,4
%A A152400 _Paul D. Hanna_, Dec 05 2008