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begins:\n  0:  1;\n  1:  4,   2;\n  2:  6,   9,   3;\n  3:  8,  20,  16,    4;\n  4: 10,  35,  45,   25,    5;\n  5: 12,  54,  96,   84,   36,    6;\n  6: 14,  77, 175,  210,  140,   49,    7;\n  7: 16, 104, 288,  440,  400,  216,   64,   8;\n  8: 18, 135, 441,  819,  945,  693,  315,  81,   9;\n  9: 20, 170, 640, 1400, 1960, 1820, 1120, 440, 100, 10;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) - (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 6 + 9*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[2, x_] := 6 + 9*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; v[3] = 6 + 9*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n - 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":"A134239","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:  4,   2;\n  2:  6,   9,   3;\n  3:  8,  20,  16,    4;\n  4: 10,  35,  45,   25,    5;\n  5: 12,  54,  96,   84,   36,    6;\n  6: 14,  77, 175,  210,  140,   49,    7;\n  7: 16, 104, 288,  440,  400,  216,   64,   8;\n  8: 18, 135, 441,  819,  945,  693,  315,  81,   9;\n  9: 20, 170, 640, 1400, 1960, 1820, 1120, 440, 100, 10;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(2*n-4)*x)*P(n-1,x) - (n-2+(2*n-4)*x+(n-2)*x^2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 6 + 9*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[2, x_] := 6 + 9*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (2*n - 4)*x)*P[n - 1, x] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 2, x])/(n - 2)]];\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] = 4 + 2*x; v[3] = 6 + 9*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (2*n - 4)*x)*v[i - 1] - (n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 2])/(n - 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-e917c954825de4ec","steps":[],"edges":[],"text":"A134239"},"headline":"A134239: A127899(unsigned) * A007318.","chain":null}