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A115085 Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n-1 from T(n-1,k) to T(n-1,n-1) with the vector of terms in column k+1 from T(k+1,k+1) to T(n,k+1): T(n,k) = Sum_{j=0..n-k-1} T(n-1,j+k)*T(j+k+1,k+1) for n>k+1>0, with T(n,n) = 1 and T(n,n-1) = n (n>=1).

%I A115085 #9 Dec 21 2015 01:16:41
%S A115085 1,1,1,3,2,1,12,5,3,1,58,21,7,4,1,321,102,32,9,5,1,1963,579,158,45,11,
%T A115085 6,1,13053,3601,933,226,60,13,7,1,92946,24426,5939,1395,306,77,15,8,1,
%U A115085 702864,176858,41385,9097,1977,398,96,17,9,1,5599204,1359906,306070
%N A115085 Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n-1 from T(n-1,k) to T(n-1,n-1) with the vector of terms in column k+1 from T(k+1,k+1) to T(n,k+1): T(n,k) = Sum_{j=0..n-k-1} T(n-1,j+k)*T(j+k+1,k+1) for n>k+1>0, with T(n,n) = 1 and T(n,n-1) = n (n>=1).
%C A115085 Triangle A115080 is the dual of this triangle.
%H A115085 Paul D. Hanna, <a href="/A115085/b115085.txt">Table of n, a(n) for n = 0..405, as a flattened triangle of rows 0..27.</a>
%e A115085 T(n,k)=[T(n-1,k),T(n-1,k+1),..,T(n-1,n-1)]*[T(k+1,k+1),T(k+2,k+1),..,T(n,k+1)]:
%e A115085 12 = [3,2,1]*[1,2,5] = 3*1 + 2*2 + 1*5;
%e A115085 21 = [5,3,1]*[1,3,7] = 5*1 + 3*3 + 1*7;
%e A115085 102 = [21,7,4,1]*[1,3,7,32] = 21*1 + 7*3 + 4*7 + 1*32;
%e A115085 158 = [32,9,5,1]*[1,4,9,45] = 32*1 + 9*4 + 5*9 + 1*45.
%e A115085 Triangle begins:
%e A115085 1;
%e A115085 1, 1;
%e A115085 3, 2, 1;
%e A115085 12, 5, 3, 1;
%e A115085 58, 21, 7, 4, 1;
%e A115085 321, 102, 32, 9, 5, 1;
%e A115085 1963, 579, 158, 45, 11, 6, 1;
%e A115085 13053, 3601, 933, 226, 60, 13, 7, 1;
%e A115085 92946, 24426, 5939, 1395, 306, 77, 15, 8, 1;
%e A115085 702864, 176858, 41385, 9097, 1977, 398, 96, 17, 9, 1;
%e A115085 5599204, 1359906, 306070, 65310, 13195, 2691, 502, 117, 19, 10, 1;
%e A115085 46746501, 10996740, 2403792, 494022, 97701, 18353, 3549, 618, 140, 21, 11, 1;
%e A115085 407019340, 93136545, 19799468, 3970878, 755834, 140178, 24691, 4563, 746, 165, 23, 12, 1; ...
%o A115085 (PARI) {T(n,k)=if(n==k,1,if(n==k+1,n, sum(j=0,n-k-1,T(n-1,j+k)*T(j+k+1,k+1))))}
%o A115085 for(n=0,12,for(k=0,n, print1(T(n,k),", "));print(""))
%Y A115085 Cf. A115086 (column 0), A115087 (column 1), A115088 (column 2), A115089 (row sums); A115080 (dual triangle).
%K A115085 nonn,tabl
%O A115085 0,4
%A A115085 _Paul D. Hanna_, Jan 13 2006

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