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begins:\n  0: 1;\n  1: 1,  3;\n  2: 1,  6,   6;\n  3: 1,  9,  21,   12;\n  4: 1, 12,  45,   60,   24;\n  5: 1, 15,  78,  171,  156,    48;\n  6: 1, 18, 120,  372,  558,   384,    96;\n  7: 1, 21, 171,  690, 1473,  1656,   912,   192;\n  8: 1, 24, 231, 1152, 3225,  5160,  4608,  2112,  384;\n  9: 1, 27, 300, 1785, 6219, 13083, 16584, 12240, 4800, 768;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, x] + x*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] + x*v[i - 2]); 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":"A116412","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   3,    1;\n  2:   6,    6,     1;\n  3:  12,   21,     9,     1;\n  4:  24,   60,    45,    12,     1;\n  5:  48,  156,   171,    78,    15,    1;\n  6:  96,  384,   558,   372,   120,   18,    1;\n  7: 192,  912,  1656,  1473,   690,  171,   21,   1;\n  8: 384, 2112,  4608,  5160,  3225, 1152,  231,  24,  1;\n  9: 768, 4800, 12240, 16584, 13083, 6219, 1785, 300, 27, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+2*x)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, x] + x*P[n - 2, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] + x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A249251","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  3;\n  2: 1,  6,   6;\n  3: 1,  9,  21,   12;\n  4: 1, 12,  45,   60,   24;\n  5: 1, 15,  78,  171,  156,    48;\n  6: 1, 18, 120,  372,  558,   384,    96;\n  7: 1, 21, 171,  690, 1473,  1656,   912,   192;\n  8: 1, 24, 231, 1152, 3225,  5160,  4608,  2112,  384;\n  9: 1, 27, 300, 1785, 6219, 13083, 16584, 12240, 4800, 768;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x)*P[n - 1, x] + x*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 2*x)*v[i - 1] + x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-e7331b0d50084c15","steps":[],"edges":[],"text":"A249251"},"headline":"A249251: Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","chain":null}