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v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (6 + (3*n + 3)*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":"A257626","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:        6,          6;\n  2:       36,        108,          36;\n  3:      216,       1404,        1404,          216;\n  4:     1296,      15876,       33696,        15876,         1296;\n  5:     7776,     166212,      642492,       642492,       166212,         7776;\n  6:    46656,    1659204,    10701720,     19274760,     10701720,      1659204,        46656;\n  7:   279936,   16052580,   163263924,    481752360,    481752360,    163263924,     16052580,      279936;\n  8:  1679616,  152031492,  2344429008,  10654827804,  17343084960,  10654827804,   2344429008,   152031492,    1679616;\n  9: 10077696, 1418671908, 32237998380, 216088713252, 535926913164, 535926913164, 216088713252, 32237998380, 1418671908, 10077696;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (6+(3*n+3)*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[(6 + (3*n + 3)*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] = (6 + (3*n + 3)*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-61cf3a32da4a0e22","steps":[],"edges":[],"text":"A257626"},"headline":"A257626: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.","chain":null}