cp's OEIS Frontend

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.

%I A123305 #10 Jan 13 2025 04:10:08
%S A123305 1,1,1,2,2,1,6,8,3,1,26,44,18,4,1,156,312,144,32,5,1,1234,2776,1422,
%T A123305 336,50,6,1,12340,30312,16848,4256,650,72,7,1,150994,397728,235458,
%U A123305 63072,10050,1116,98,8,1,2204112,6151768,3827628,1076128,178900,20376,1764
%N 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.
%F A123305 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.
%e A123305 Triangle begins:
%e A123305   1;
%e A123305   1, 1;
%e A123305   2, 2, 1;
%e A123305   6, 8, 3, 1;
%e A123305   26, 44, 18, 4, 1;
%e A123305   156, 312, 144, 32, 5, 1;
%e A123305   1234, 2776, 1422, 336, 50, 6, 1;
%e A123305   12340, 30312, 16848, 4256, 650, 72, 7, 1;
%e A123305   150994, 397728, 235458, 63072, 10050, 1116, 98, 8, 1;
%e A123305   ...
%e A123305 Matrix square starts:
%e A123305   1;
%e A123305   2, 1;
%e A123305   6, 4, 1;
%e A123305   26, 22, 6, 1;
%e A123305   156, 156, 48, 8, 1;
%e A123305   1234, 1388, 474, 84, 10, 1;
%e A123305   12340, 15156, 5616, 1064, 130, 12, 1;
%e A123305   ...
%e A123305 Note that (column k of T shifted) = (k+1)*(column k of T^2):
%e A123305   k=1: [2,8,44,312,2776,...] = 2*[1,4,22,156,1388,...];
%e A123305   k=2: [3,18,144,1422,16848,....] = 3*[1,6,48,474,5616,...].
%o A123305 (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)); ))}
%Y A123305 Columns: A123306, A123307, A123308, A123309.
%Y A123305 Cf. A123310 (row sums), A123311 (central terms).
%K A123305 nonn,tabl
%O A123305 0,4
%A A123305 _Paul D. Hanna_, Sep 24 2006