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begins:\n  0: 1;\n  1: 0, 2;\n  2: 0, 2,  3;\n  3: 0, 2,  8,   4;\n  4: 0, 2, 15,  20,    5;\n  5: 0, 2, 24,  60,   40,    6;\n  6: 0, 2, 35, 140,  175,   70,    7;\n  7: 0, 2, 48, 280,  560,  420,  112,    8;\n  8: 0, 2, 63, 504, 1470, 1764,  882,  168,   9;\n  9: 0, 2, 80, 840, 3360, 5880, 4704, 1680, 240, 10;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+3*n-2)*P(n,x) = (4*n^2-4+(4*n^2+2)*x)*P(n-1,x) - (2*n^2-3*n-2-(4*n^2-6*n-4)*x+(2*n^2-3*n-2)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 4 + (4*n^2 + 2)*x)*P[n - 1, x] - (2*n^2 - 3*n - 2 - (4*n^2 - 6*n - 4)*x + (2*n^2 - 3*n - 2)*x^2)*P[n - 2, x])/(2*n^2 + 3*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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 4 + (4*n^2 + 2)*x)*v[i - 1] - (2*n^2 - 3*n - 2 - (4*n^2 - 6*n - 4)*x + (2*n^2 - 3*n - 2)*x^2)*v[i - 2])/(2*n^2 + 3*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":"A108838","target":"reversed_triangle","first_row":1,"oeis_offset":2,"reversed":true,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   2:  2;\n   3:  3,   2;\n   4:  4,   8,    2;\n   5:  5,  20,   15,     2;\n   6:  6,  40,   60,    24,     2;\n   7:  7,  70,  175,   140,    35,     2;\n   8:  8, 112,  420,   560,   280,    48,    2;\n   9:  9, 168,  882,  1764,  1470,   504,   63,    2;\n  10: 10, 240, 1680,  4704,  5880,  3360,  840,   80,  2;\n  11: 11, 330, 2970, 11088, 19404, 16632, 6930, 1320, 99, 2;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where (2*n^2-n-3)*P(n,x) = (4*n^2-8*n+(4*n^2-8*n+6)*x)*P(n-1,x) - (2*n^2-7*n+3-(4*n^2-14*n+6)*x+(2*n^2-7*n+3)*x^2)*P(n-2,x) for n >= 4, with P(2,x) = 2*x, P(3,x) = 2*x + 3*x^2.","mathematica":"P[2, x_] := 2*x;\nP[3, x_] := 2*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 8*n + (4*n^2 - 8*n + 6)*x)*P[n - 1, x] - (2*n^2 - 7*n + 3 - (4*n^2 - 14*n + 6)*x + (2*n^2 - 7*n + 3)*x^2)*P[n - 2, x])/(2*n^2 - n - 3)]];\nTable[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n - 1]], {n, 2, 11}] // Flatten","pari":"P(N) = {my(v = vector(N - 1)); v[1] = 2*x; v[2] = 2*x + 3*x^2; for(n = 4, N, my(i = n - 1); v[i] = ((4*n^2 - 8*n + (4*n^2 - 8*n + 6)*x)*v[i - 1] - (2*n^2 - 7*n + 3 - (4*n^2 - 14*n + 6)*x + (2*n^2 - 7*n + 3)*x^2)*v[i - 2])/(2*n^2 - n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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(11)","reading":{"first_row":1,"offset":2,"reversed":true,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A281260","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0, 2;\n   3: 0, 2,  3;\n   4: 0, 2,  8,   4;\n   5: 0, 2, 15,  20,    5;\n   6: 0, 2, 24,  60,   40,    6;\n   7: 0, 2, 35, 140,  175,   70,    7;\n   8: 0, 2, 48, 280,  560,  420,  112,    8;\n   9: 0, 2, 63, 504, 1470, 1764,  882,  168,   9;\n  10: 0, 2, 80, 840, 3360, 5880, 4704, 1680, 240, 10;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-n-3)*P(n,x) = (4*n^2-8*n+(4*n^2-8*n+6)*x)*P(n-1,x) - (2*n^2-7*n+3-(4*n^2-14*n+6)*x+(2*n^2-7*n+3)*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 8*n + (4*n^2 - 8*n + 6)*x)*P[n - 1, x] - (2*n^2 - 7*n + 3 - (4*n^2 - 14*n + 6)*x + (2*n^2 - 7*n + 3)*x^2)*P[n - 2, x])/(2*n^2 - n - 3)]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 8*n + (4*n^2 - 8*n + 6)*x)*v[i - 1] - (2*n^2 - 7*n + 3 - (4*n^2 - 14*n + 6)*x + (2*n^2 - 7*n + 3)*x^2)*v[i - 2])/(2*n^2 - n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-8a65e165240c40e0","steps":[],"edges":[],"text":"A281260"},"headline":"A281260: Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by rows.","chain":null}