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%I #13 May 02 2023 01:50:36
%S 1,1,1,3,1,1,18,4,1,1,170,30,5,1,1,2220,335,45,6,1,1,37149,4984,581,
%T 63,7,1,1,758814,92652,9730,924,84,8,1,1,18301950,2065146,199692,
%U 17226,1380,108,9,1,1,508907970,53636520,4843125,387567,28365,1965,135,10,1,1
%N Triangular matrix T, read by rows, where row n of T equals row (n-1) of T^(n+1) with an appended '1'.
%C Related to the number of subpartitions of a partition as defined in A115728; for examples involving column k of successive matrix powers, see A121430, A121431, A121432 and A121433. Essentially the same as triangle A101479, but this form best illustrates the nice properties of this triangle.
%H Alois P. Heinz, <a href="/A121412/b121412.txt">Rows n = 0..45, flattened</a>
%F G.f.: Column k of successive powers of T satisfy the amazing relation given by: 1 = Sum_{n>=0} (1-x)^(n+1) * x^(n(n+1)/2 + k*n) * Sum_{j=0..n+k} [T^(j+1)](n+k,k) * x^j.
%e Triangle T begins:
%e 1;
%e 1, 1;
%e 3, 1, 1;
%e 18, 4, 1, 1;
%e 170, 30, 5, 1, 1;
%e 2220, 335, 45, 6, 1, 1;
%e 37149, 4984, 581, 63, 7, 1, 1;
%e 758814, 92652, 9730, 924, 84, 8, 1, 1;
%e 18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1;
%e 508907970, 53636520, 4843125, 387567, 28365, 1965, 135, 10, 1, 1;
%e To get row 4 of T, append '1' to row 3 of matrix power T^5:
%e 1;
%e 5, 1;
%e 25, 5, 1;
%e 170, 30, 5, 1; ...
%e To get row 5 of T, append '1' to row 4 of matrix power T^6:
%e 1;
%e 6, 1;
%e 33, 6, 1;
%e 233, 39, 6, 1;
%e 2220, 335, 45, 6, 1; ...
%e Likewise, get row n of T by appending '1' to row (n-1) of T^(n+1).
%t T[n_, k_] := Module[{A = {{1}}, B}, Do[B = Array[0&, {m, m}]; Do[Do[B[[i, j]] = If[j == i, 1, MatrixPower[A, i][[i-1, j]]], {j, 1, i}], {i, 1, m}]; A = B, {m, 1, n+1}]; A[[n+1, k+1]]];
%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 03 2019 *)
%o (PARI) {T(n, k) = my(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j]=(A^i)[i-1, j]); )); A=B); return((A^1)[n+1, k+1])}
%o for(n=0,12, for(k=0,n, print1( T(n,k),", "));print(""))
%Y Cf. A121416 (T^2), A121420 (T^3), columns: A121413, A121414, A121415; related tables: A121424, A121426, A121428; related subpartitions: A121430, A121431, A121432, A121433.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Jul 30 2006
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