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 A121412 Triangular matrix T, read by rows, where row n of T equals row (n-1) of T^(n+1) with an appended '1'. 30
 1, 1, 1, 3, 1, 1, 18, 4, 1, 1, 170, 30, 5, 1, 1, 2220, 335, 45, 6, 1, 1, 37149, 4984, 581, 63, 7, 1, 1, 758814, 92652, 9730, 924, 84, 8, 1, 1, 18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1, 508907970, 53636520, 4843125, 387567, 28365, 1965, 135, 10, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. LINKS Alois P. Heinz, Rows n = 0..45, flattened FORMULA 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. EXAMPLE Triangle T begins: 1; 1, 1; 3, 1, 1; 18, 4, 1, 1; 170, 30, 5, 1, 1; 2220, 335, 45, 6, 1, 1; 37149, 4984, 581, 63, 7, 1, 1; 758814, 92652, 9730, 924, 84, 8, 1, 1; 18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1; 508907970, 53636520, 4843125, 387567, 28365, 1965, 135, 10, 1, 1; To get row 4 of T, append '1' to row 3 of matrix power T^5: 1; 5, 1; 25, 5, 1; 170, 30, 5, 1; ... To get row 5 of T, append '1' to row 4 of matrix power T^6: 1; 6, 1; 33, 6, 1; 233, 39, 6, 1; 2220, 335, 45, 6, 1; ... Likewise, get row n of T by appending '1' to row (n-1) of T^(n+1). MATHEMATICA 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]]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 03 2019 *) PROG (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])} for(n=0, 12, for(k=0, n, print1( T(n, k), ", ")); print("")) CROSSREFS Cf. A121416 (T^2), A121420 (T^3), columns: A121413, A121414, A121415; related tables: A121424, A121426, A121428; related subpartitions: A121430, A121431, A121432, A121433. Sequence in context: A333560 A176293 A176339 * A212855 A016561 A111382 Adjacent sequences: A121409 A121410 A121411 * A121413 A121414 A121415 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Jul 30 2006 STATUS approved

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Last modified May 29 07:06 EDT 2024. Contains 372926 sequences. (Running on oeis4.)