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v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A055151","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1;\n  2: 1,  1;\n  3: 1,  3;\n  4: 1,  6,   2;\n  5: 1, 10,  10;\n  6: 1, 15,  30,   5;\n  7: 1, 21,  70,  35;\n  8: 1, 28, 140, 140,  14;\n  9: 1, 36, 252, 420, 126;\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+1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*P[n - 2, x])/(n + 2)]];\nTable[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*v[i - 2])/(n + 2)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":"low","regular_triangle":false}},{"oeis_id":"A080159","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: 1,  0;\n  2: 1,  1,   0;\n  3: 1,  3,   0,   0;\n  4: 1,  6,   2,   0,   0;\n  5: 1, 10,  10,   0,   0, 0;\n  6: 1, 15,  30,   5,   0, 0, 0;\n  7: 1, 21,  70,  35,   0, 0, 0, 0;\n  8: 1, 28, 140, 140,  14, 0, 0, 0, 0;\n  9: 1, 36, 252, 420, 126, 0, 0, 0, 0, 0;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n+2)*P(n,x) = (2*n+1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*x^2)*P[n - 2, x])/(n + 2)]];\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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*x^2)*v[i - 2])/(n + 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":"A107131","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, 1, 1;\n  3: 0, 0, 3,  1;\n  4: 0, 0, 2,  6,  1;\n  5: 0, 0, 0, 10, 10,   1;\n  6: 0, 0, 0,  5, 30,  15,   1;\n  7: 0, 0, 0,  0, 35,  70,  21,   1;\n  8: 0, 0, 0,  0, 14, 140, 140,  28,  1;\n  9: 0, 0, 0,  0,  0, 126, 420, 252, 36, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (2*n+1)*x*P(n-1,x) + ((4*n-4)*x-(n-1)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 1)*x*P[n - 1, x] + ((4*n - 4)*x - (n - 1)*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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1)*x*v[i - 1] + ((4*n - 4)*x - (n - 1)*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-de4284a2d5d7d0a6","steps":[],"edges":[],"text":"A107131"},"headline":"A107131: A Motzkin related triangle.","chain":null}