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x], x]];\nTable[CoefficientList[P[n, x], x], {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 + (7*n - 4)*x)*v[i - 1] + (7*x - 7*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":"A257627","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,        60,           9;\n  3:    27,       753,         753,            27;\n  4:    81,      8178,       25602,          8178,            81;\n  5:   243,     84291,      631506,        631506,         84291,           243;\n  6:   729,    852144,    13348623,      30312288,      13348623,        852144,           729;\n  7:  2187,   8554245,   259308063,    1141302225,    1141302225,     259308063,       8554245,        2187;\n  8:  6561,  85656174,  4793178096,   37244959794,   70760737950,   37244959794,    4793178096,    85656174,      6561;\n  9: 19683, 856948839, 85938148680, 1109572049376, 3608891348622, 3608891348622, 1109572049376, 85938148680, 856948839, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(7*n-4)*x)*P(n-1,x) + (7*x-7*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 + (7*n - 4)*x)*P[n - 1, x] + (7*x - 7*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 + (7*n - 4)*x)*v[i - 1] + (7*x - 7*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":"A257627","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,        60,           9;\n  3:    27,       753,         753,            27;\n  4:    81,      8178,       25602,          8178,            81;\n  5:   243,     84291,      631506,        631506,         84291,           243;\n  6:   729,    852144,    13348623,      30312288,      13348623,        852144,           729;\n  7:  2187,   8554245,   259308063,    1141302225,    1141302225,     259308063,       8554245,        2187;\n  8:  6561,  85656174,  4793178096,   37244959794,   70760737950,   37244959794,    4793178096,    85656174,      6561;\n  9: 19683, 856948839, 85938148680, 1109572049376, 3608891348622, 3608891348622, 1109572049376, 85938148680, 856948839, 19683;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+(7*n-4)*x)*P(n-1,x) + (7*x-7*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 + (7*n - 4)*x)*P[n - 1, x] + (7*x - 7*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 + (7*n - 4)*x)*v[i - 1] + (7*x - 7*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-64714e0ad28cd8ed","steps":[],"edges":[],"text":"A257627"},"headline":"A257627: 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(x) = 7*x + 3.","chain":null}