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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 - (4*n - 4)*x + (4*n - 4)*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":"A114656","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:   4,     6,     1;\n   4:   8,    24,    12,      1;\n   5:  16,    80,    80,     20,      1;\n   6:  32,   240,   400,    200,     30,     1;\n   7:  64,   672,  1680,   1400,    420,    42,     1;\n   8: 128,  1792,  6272,   7840,   3920,   784,    56,    1;\n   9: 256,  4608, 21504,  37632,  28224,  9408,  1344,   72,  1;\n  10: 512, 11520, 69120, 161280, 169344, 84672, 20160, 2160, 90, 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-(4*n-8)*x+(4*n-8)*x^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 - (4*n - 8)*x + (4*n - 8)*x^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 - (4*n - 8)*x + (4*n - 8)*x^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}},{"oeis_id":"A114687","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: 1,  2;\n   3: 1,  6,    4;\n   4: 1, 12,   24,     8;\n   5: 1, 20,   80,    80,    16;\n   6: 1, 30,  200,   400,   240,     32;\n   7: 1, 42,  420,  1400,  1680,    672,     64;\n   8: 1, 56,  784,  3920,  7840,   6272,   1792,   128;\n   9: 1, 72, 1344,  9408, 28224,  37632,  21504,  4608,   256;\n  10: 1, 90, 2160, 20160, 84672, 169344, 161280, 69120, 11520, 512;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (2*n-1+(4*n-2)*x)*P(n-1,x) - (n-2-(4*n-8)*x+(4*n-8)*x^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 - (4*n - 8)*x + (4*n - 8)*x^2)*P[n - 2, x])/(n + 1)]];\nTable[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 - (4*n - 8)*x + (4*n - 8)*x^2)*v[i - 2])/(n + 1)); 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-8ce61d302cec295b","steps":[],"edges":[],"text":"A114687"},"headline":"A114687: Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","chain":null}