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A078123 Square of infinite lower triangular matrix A078122.

%I A078123 #13 Mar 06 2015 09:55:16
%S A078123 1,2,1,5,6,1,23,51,18,1,239,861,477,54,1,5828,32856,25263,4347,162,1,
%T A078123 342383,3013980,3016107,699813,39285,486,1,50110484,690729981,
%U A078123 865184724,253656252,19053063,354051,1458,1,18757984046,406279238154
%N A078123 Square of infinite lower triangular matrix A078122.
%H A078123 Alois P. Heinz, <a href="/A078123/b078123.txt">Rows n = 0..60, flattened</a>
%F A078123 M(1, j) = A078125(j), M(j+1, j)=2*3^j.
%e A078123 Square of A078122 = A078123 as can be seen by 4 X 4 submatrix:
%e A078123 [1,_0,_0,0]^2=[_1,_0,_0,_0]
%e A078123 [1,_1,_0,0]___[_2,_1,_0,_0]
%e A078123 [1,_3,_1,0]___[_5,_6,_1,_0]
%e A078123 [1,12,_9,1]___[23,51,18,_1]
%p A078123 S:= proc(i, j) option remember;
%p A078123        add(M(i, k)*M(k, j), k=0..i)
%p A078123     end:
%p A078123 M:= proc(i, j) option remember; `if`(j=0 or i=j, 1,
%p A078123        add(S(i-1, k)*M(k, j-1), k=0..i-1))
%p A078123     end:
%p A078123 seq(seq(S(n,k), k=0..n), n=0..10);  # _Alois P. Heinz_, Feb 27 2015
%t A078123 S[i_, j_] := S[i, j] = Sum[M[i, k]*M[k, j], {k, 0, i}]; M[i_, j_] := M[i, j] = If[j == 0 || i == j, 1, Sum[S[i-1, k]*M[k, j-1], {k, 0, i-1}]]; Table[Table[S[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Mar 06 2015, after _Alois P. Heinz_ *)
%Y A078123 Cf. A078121, A078122, A078124, A078125.
%K A078123 nonn,tabl
%O A078123 0,2
%A A078123 _Paul D. Hanna_, Nov 18 2002