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begins:\n  0: 1;\n  1: 0, 1;\n  2: 0, 2,   1;\n  3: 0, 3,   4,   1;\n  4: 0, 4,  10,   6,   1;\n  5: 0, 5,  20,  21,   8,   1;\n  6: 0, 6,  35,  56,  36,  10,   1;\n  7: 0, 7,  56, 126, 120,  55,  12,   1;\n  8: 0, 8,  84, 252, 330, 220,  78,  14,  1;\n  9: 0, 9, 120, 462, 792, 715, 364, 105, 16, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(n-2)*x)*P(n-1,x) - (n-2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = x, P(2,x) = 2*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[2, x_] := 2*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (n - 2)*x)*P[n - 1, x] - (n - 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] = x; v[3] = 2*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (n - 2)*x)*v[i - 1] - (n - 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":"A078812","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  2,   1;\n  2:  3,   4,   1;\n  3:  4,  10,   6,    1;\n  4:  5,  20,  21,    8,    1;\n  5:  6,  35,  56,   36,   10,    1;\n  6:  7,  56, 126,  120,   55,   12,   1;\n  7:  8,  84, 252,  330,  220,   78,  14,   1;\n  8:  9, 120, 462,  792,  715,  364, 105,  16,  1;\n  9: 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) - (n-1)*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = 2*x + x^2.","mathematica":"P[0, x_] := x;\nP[1, x_] := 2*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x] - (n - 1)*P[n - 2, x])/(n - 1)]];\nTable[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 2*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1] - (n - 1)*v[i - 2])/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":1,"offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A128908","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: 0, 1;\n  2: 0, 2,   1;\n  3: 0, 3,   4,   1;\n  4: 0, 4,  10,   6,   1;\n  5: 0, 5,  20,  21,   8,   1;\n  6: 0, 6,  35,  56,  36,  10,   1;\n  7: 0, 7,  56, 126, 120,  55,  12,   1;\n  8: 0, 8,  84, 252, 330, 220,  78,  14,  1;\n  9: 0, 9, 120, 462, 792, 715, 364, 105, 16, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (2*n-4+(n-2)*x)*P(n-1,x) - (n-2)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = x, P(2,x) = 2*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[2, x_] := 2*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (n - 2)*x)*P[n - 1, x] - (n - 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] = x; v[3] = 2*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n - 4 + (n - 2)*x)*v[i - 1] - (n - 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}},{"oeis_id":"A172431","target":"reversed_triangle","first_row":1,"oeis_offset":1,"reversed":true,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  2;\n   3: 1,  4,   3;\n   4: 1,  6,  10,   4;\n   5: 1,  8,  21,  20,    5;\n   6: 1, 10,  36,  56,   35,    6;\n   7: 1, 12,  55, 120,  126,   56,    7;\n   8: 1, 14,  78, 220,  330,  252,   84,   8;\n   9: 1, 16, 105, 364,  715,  792,  462, 120,   9;\n  10: 1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where (n-2)*P(n,x) = (2*n-4+(n-2)*x)*P(n-1,x) - (n-2)*P(n-2,x) for n >= 3, with P(1,x) = x, P(2,x) = 2*x + x^2.","mathematica":"P[1, x_] := x;\nP[2, x_] := 2*x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 4 + (n - 2)*x)*P[n - 1, x] - (n - 2)*P[n - 2, x])/(n - 2)]];\nTable[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = x; v[2] = 2*x + x^2; for(n = 3, N, my(i = n); v[i] = ((2*n - 4 + (n - 2)*x)*v[i - 1] - (n - 2)*v[i - 2])/(n - 2)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":true,"trim_zeros":"low","regular_triangle":true}}],"derivation":{"root":"n-83d230649c15eed8","steps":[],"edges":[],"text":"A128908"},"headline":"A128908: Riordan array (1, x/(1-x)^2).","chain":null}