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begins:\n  0: 1;\n  1: 1,  2;\n  2: 1,  5,   5;\n  3: 1,  9,  21,   14;\n  4: 1, 14,  56,   84,    42;\n  5: 1, 20, 120,  300,   330,   132;\n  6: 1, 27, 225,  825,  1485,  1287,    429;\n  7: 1, 35, 385, 1925,  5005,  7007,   5005,   1430;\n  8: 1, 44, 616, 4004, 14014, 28028,  32032,  19448,  4862;\n  9: 1, 54, 936, 7644, 34398, 91728, 148512, 143208, 75582, 16796;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (2*n+1+(4*n+2)*x)*P(n-1,x) - (n-1)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + (4*n + 2)*x)*P[n - 1, x] - (n - 1)*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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 1 + (4*n + 2)*x)*v[i - 1] - (n - 1)*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":"A033282","target":"triangle","first_row":0,"oeis_offset":3,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   3: 1;\n   4: 1,  2;\n   5: 1,  5,   5;\n   6: 1,  9,  21,   14;\n   7: 1, 14,  56,   84,    42;\n   8: 1, 20, 120,  300,   330,   132;\n   9: 1, 27, 225,  825,  1485,  1287,    429;\n  10: 1, 35, 385, 1925,  5005,  7007,   5005,   1430;\n  11: 1, 44, 616, 4004, 14014, 28028,  32032,  19448,  4862;\n  12: 1, 54, 936, 7644, 34398, 91728, 148512, 143208, 75582, 16796;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-5+(4*n-10)*x)*P(n-1,x) - (n-4)*P(n-2,x) for n >= 5, with P(3,x) = 1, P(4,x) = 1 + 2*x.","mathematica":"P[3, x_] := 1;\nP[4, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 5 + (4*n - 10)*x)*P[n - 1, x] - (n - 4)*P[n - 2, x])/(n - 1)]];\nTable[PadRight[CoefficientList[P[n, x], x], n - 2], {n, 3, 12}] // Flatten","pari":"P(N) = {my(v = vector(N - 2)); v[1] = 1; v[2] = 1 + 2*x; for(n = 5, N, my(i = n - 2); v[i] = ((2*n - 5 + (4*n - 10)*x)*v[i - 1] - (n - 4)*v[i - 2])/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 2 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 2, v[i])))};\nT(12)","reading":{"first_row":0,"offset":3,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A126216","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     2,     1;\n   3:     5,     5,      1;\n   4:    14,    21,      9,      1;\n   5:    42,    84,     56,     14,     1;\n   6:   132,   330,    300,    120,    20,     1;\n   7:   429,  1287,   1485,    825,   225,    27,    1;\n   8:  1430,  5005,   7007,   5005,  1925,   385,   35,   1;\n   9:  4862, 19448,  32032,  28028, 14014,  4004,  616,  44,  1;\n  10: 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n+1)*P(n,x) = (2*n-1+(4*n-2)*x)*P(n-1,x) - (n-2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (4*n - 2)*x)*P[n - 1, x] - (n - 2)*P[n - 2, x])/(n + 1)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = ((2*n - 1 + (4*n - 2)*x)*v[i - 1] - (n - 2)*v[i - 2])/(n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); 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":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-66e2c3448245b0b7","steps":[],"edges":[],"text":"A033282"},"headline":"A033282: Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","chain":null}