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A125278 Convolution triangle of A030266, which shifts left under self-COMPOSE.

%I A125278 #5 Jun 14 2017 00:14:05
%S A125278 1,1,1,2,2,1,6,5,3,1,23,16,9,4,1,104,62,31,14,5,1,531,278,123,52,20,6,
%T A125278 1,2982,1398,552,213,80,27,7,1,18109,7718,2750,964,340,116,35,8,1,
%U A125278 117545,46083,14976,4784,1561,513,161,44,9,1,808764,294392,88083,25792,7755
%N A125278 Convolution triangle of A030266, which shifts left under self-COMPOSE.
%C A125278 Column 0 of matrix square T^2 equals column 0 of T shift left. Central terms are T(2*n,n) = (n+1)*A125279(n).
%F A125278 T(0,0) = 1 ; for n>0: T(n,0) = Sum_{j=0..n-1} T(j,0)*T(n-1,j) = A030266(n) (self-COMPOSE); for k>0: T(n,k) = Sum_{j=0..n-k} T(j,0)*T(n-1-j,k-1) (self-convolutions of A030266).
%e A125278 Triangle begins:
%e A125278 1;
%e A125278 1, 1;
%e A125278 2, 2, 1;
%e A125278 6, 5, 3, 1;
%e A125278 23, 16, 9, 4, 1;
%e A125278 104, 62, 31, 14, 5, 1;
%e A125278 531, 278, 123, 52, 20, 6, 1;
%e A125278 2982, 1398, 552, 213, 80, 27, 7, 1;
%e A125278 18109, 7718, 2750, 964, 340, 116, 35, 8, 1;
%e A125278 117545, 46083, 14976, 4784, 1561, 513, 161, 44, 9, 1;
%e A125278 808764, 294392, 88083, 25792, 7755, 2400, 742, 216, 54, 10, 1; ...
%e A125278 Matrix square T^2 begins:
%e A125278 1;
%e A125278 2, 1;
%e A125278 6, 4, 1;
%e A125278 23, 16, 6, 1;
%e A125278 104, 70, 30, 8, 1;
%e A125278 531, 336, 149, 48, 10, 1; ...
%e A125278 which is also a convolution triangle.
%o A125278 (PARI) T(n,k)=if(n<k || k<0,0,if(n==k,1,if(k>0,sum(j=0,n-k,T(j,0)*T(n-1-j,k-1)), sum(j=0,n-1,T(j,0)*T(n-1,j)))))
%Y A125278 Cf. A030266, A125279, A125280 (row sums).
%K A125278 nonn,tabl
%O A125278 0,4
%A A125278 _Paul D. Hanna_, Nov 26 2006