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%I #15 Aug 09 2018 04:49:31
%S 1,1,1,6,2,1,46,10,3,1,416,72,16,4,1,4256,632,116,24,5,1,48096,6352,
%T 1016,184,34,6,1,591536,70912,10176,1664,282,46,7,1,7840576,864192,
%U 113216,17024,2696,416,60,8,1,111226816,11371072,1375456,192384,28792,4256,592,76,9,1
%N Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+4 of T), or [T^p](m,0) = p*T(p+m,p+4) for all m>=1 and p>=-4.
%C Column 0 equals A111531 (related to log of factorial series). Column 4 (A111557) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A111560.
%H Paul Barry, <a href="https://arxiv.org/abs/1804.06801">A note on number triangles that are almost their own production matrix</a>, arXiv:1804.06801 [math.CO], 2018.
%F T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j+3, 3)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+4, 3) = 4*T(n+1, 0), T(n+5, 5) = T(n+1, 0), for n>=0.
%e SHIFT_LEFT(column 0 of T^-4) = -4*(column 0 of T);
%e SHIFT_LEFT(column 0 of T^-3) = -3*(column 1 of T);
%e SHIFT_LEFT(column 0 of T^-2) = -2*(column 2 of T);
%e SHIFT_LEFT(column 0 of T^-1) = -1*(column 3 of T);
%e SHIFT_LEFT(column 0 of log(T)) = column 4 of T;
%e SHIFT_LEFT(column 0 of T^1) = 1*(column 5 of T);
%e where SHIFT_LEFT of column sequence shifts 1 place left.
%e Triangle T begins:
%e 1;
%e 1,1;
%e 6,2,1;
%e 46,10,3,1;
%e 416,72,16,4,1;
%e 4256,632,116,24,5,1;
%e 48096,6352,1016,184,34,6,1;
%e 591536,70912,10176,1664,282,46,7,1;
%e 7840576,864192,113216,17024,2696,416,60,8,1; ...
%e After initial term, column 3 is 4 times column 0.
%e Matrix inverse T^-1 = A111559 starts:
%e 1;
%e -1,1;
%e -4,-2,1;
%e -24,-4,-3,1;
%e -184,-24,-4,-4,1;
%e -1664,-184,-24,-4,-5,1;
%e -17024,-1664,-184,-24,-4,-6,1; ...
%e where columns are all equal after initial terms;
%e compare columns of T^-1 to column 3 of T.
%e Matrix logarithm log(T) = A111560 is:
%e 0;
%e 1,0;
%e 5,2,0;
%e 34,7,3,0;
%e 282,44,10,4,0;
%e 2696,354,60,14,5,0;
%e 28792,3328,470,84,19,6,0; ...
%e compare column 0 of log(T) to column 4 of T.
%t T[n_, k_] := T[n, k] = If[n<k || k<0, 0, If[n == k, 1, If[n == k + 1, n, k T[n, k + 1] + Sum[T[j + 3, 3] T[n, j + k + 1], {j, 0, n - k - 1}]]]];
%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 09 2018, from PARI *)
%o (PARI) T(n,k)=if(n<k || k<0,0,if(n==k,1,if(n==k+1,n, k*T(n,k+1)+sum(j=0,n-k-1,T(j+3,3)*T(n,j+k+1)))))
%Y Cf. A111531 (column 0), A111554 (column 1), A111555 (column 2), A111556 (column 3), A111557 (column 4), A111558 (row sums), A111559 (matrix inverse), A111560 (matrix log); related tables: A111528, A104980, A111536, A111544.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Aug 07 2005
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