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vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3 + 3*n*x)*v[i - 1] + (3*x - 3*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":"A257620","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,       36,         9;\n  3:    27,      297,       297,         27;\n  4:    81,     2106,      5346,       2106,          81;\n  5:   243,    13851,     73386,      73386,       13851,         243;\n  6:   729,    87480,    868239,    1761264,      868239,       87480,        729;\n  7:  2187,   540189,   9388791,   34158753,    34158753,     9388791,     540189,      2187;\n  8:  6561,  3293622,  95843088,  578903274,  1024762590,   578903274,   95843088,   3293622,     6561;\n  9: 19683, 19938879, 941634720, 8959544136, 25791697782, 25791697782, 8959544136, 941634720, 19938879, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+3*n*x)*P(n-1,x) + (3*x-3*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 + 3*n*x)*P[n - 1, x] + (3*x - 3*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 + 3*n*x)*v[i - 1] + (3*x - 3*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":"A257620","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,       36,         9;\n  3:    27,      297,       297,         27;\n  4:    81,     2106,      5346,       2106,          81;\n  5:   243,    13851,     73386,      73386,       13851,         243;\n  6:   729,    87480,    868239,    1761264,      868239,       87480,        729;\n  7:  2187,   540189,   9388791,   34158753,    34158753,     9388791,     540189,      2187;\n  8:  6561,  3293622,  95843088,  578903274,  1024762590,   578903274,   95843088,   3293622,     6561;\n  9: 19683, 19938879, 941634720, 8959544136, 25791697782, 25791697782, 8959544136, 941634720, 19938879, 19683;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+3*n*x)*P(n-1,x) + (3*x-3*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 + 3*n*x)*P[n - 1, x] + (3*x - 3*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 + 3*n*x)*v[i - 1] + (3*x - 3*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-dfedc6a36c97eba4","steps":[],"edges":[],"text":"A257620"},"headline":"A257620: Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.","chain":null}