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v[1] = 1; v[2] = 3; v[3] = 9; for(n = 3, N, my(i = n + 1); v[i] = 3*v[i - 1] + x*v[i - 3]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A317497","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     3;\n  2:     9;\n  3:    27,    1;\n  4:    81,    6;\n  5:   243,   27;\n  6:   729,  108,   1;\n  7:  2187,  405,   9;\n  8:  6561, 1458,  54;\n  9: 19683, 5103, 270, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 3*P(n-1,x) + x*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3, P(2,x) = 9.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[2, x_] := 9;\nP[n_, x_] := P[n, x] = Expand[3*P[n - 1, x] + x*P[n - 3, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; v[3] = 9; for(n = 3, N, my(i = n + 1); v[i] = 3*v[i - 1] + x*v[i - 3]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-0d55e0bb3c384646","steps":["x \u21a6 (1/27)x"],"edges":["e-b8d37b5fe37403cf"],"text":"A102547 \u2192 x \u21a6 (1/27)x"},"headline":"A317497: Triangle T(n,k) = 3*T(n-1,k) +  T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.","chain":"A102547 \u2192 x \u21a6 (1/27)x"}