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v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + (5*n - 2)*x)*v[i - 1] + (5*x - 5*x^2)*deriv(v[i - 1], x)); 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":"A257623","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:     3,         3;\n  2:     9,        48,           9;\n  3:    27,       501,         501,           27;\n  4:    81,      4494,       13026,         4494,           81;\n  5:   243,     37815,      250230,       250230,        37815,          243;\n  6:   729,    309324,     4122735,      9008280,      4122735,       309324,          729;\n  7:  2187,   2498649,    62256627,    256971945,    256971945,     62256627,      2498649,        2187;\n  8:  6561,  20072298,   891791568,   6368680566,  11820709470,   6368680566,    891791568,    20072298,      6561;\n  9: 19683, 160860507, 12356037708, 144065371932, 450199373658, 450199373658, 144065371932, 12356037708, 160860507, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(5*n-2)*x)*P(n-1,x) + (5*x-5*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + (5*n - 2)*x)*P[n - 1, x] + (5*x - 5*x^2)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + (5*n - 2)*x)*v[i - 1] + (5*x - 5*x^2)*deriv(v[i - 1], x)); 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":"A257623","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:     3,         3;\n  2:     9,        48,           9;\n  3:    27,       501,         501,           27;\n  4:    81,      4494,       13026,         4494,           81;\n  5:   243,     37815,      250230,       250230,        37815,          243;\n  6:   729,    309324,     4122735,      9008280,      4122735,       309324,          729;\n  7:  2187,   2498649,    62256627,    256971945,    256971945,     62256627,      2498649,        2187;\n  8:  6561,  20072298,   891791568,   6368680566,  11820709470,   6368680566,    891791568,    20072298,      6561;\n  9: 19683, 160860507, 12356037708, 144065371932, 450199373658, 450199373658, 144065371932, 12356037708, 160860507, 19683;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+(5*n-2)*x)*P(n-1,x) + (5*x-5*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + (5*n - 2)*x)*P[n - 1, x] + (5*x - 5*x^2)*D[P[n - 1, x], x]];\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; for(n = 1, N, my(i = n + 1); v[i] = (3 + (5*n - 2)*x)*v[i - 1] + (5*x - 5*x^2)*deriv(v[i - 1], x)); 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}}],"derivation":{"root":"n-4155b181fd90d601","steps":[],"edges":[],"text":"A257623"},"headline":"A257623: Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.","chain":null}