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begins:\n  0:     1;\n  1:     3,         3;\n  2:     9,        54,           9;\n  3:    27,       621,         621,           27;\n  4:    81,      6156,       18630,         6156,            81;\n  5:   243,     57591,      408726,       408726,         57591,           243;\n  6:   729,    526338,     7685847,     17166492,       7685847,        526338,          729;\n  7:  2187,   4765473,   132656859,    568014201,     568014201,     132656859,      4765473,        2187;\n  8:  6561,  42987672,  2175706332,  16305974568,   30672766854,   16305974568,   2175706332,    42987672,      6561;\n  9: 19683, 387223659, 34570040220, 427278012876, 1366261865802, 1366261865802, 427278012876, 34570040220, 387223659, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(6*n-3)*x)*P(n-1,x) + (6*x-6*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 + (6*n - 3)*x)*P[n - 1, x] + (6*x - 6*x^2)*D[P[n - 1, x], 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7685847,        526338,          729;\n  7:  2187,   4765473,   132656859,    568014201,     568014201,     132656859,      4765473,        2187;\n  8:  6561,  42987672,  2175706332,  16305974568,   30672766854,   16305974568,   2175706332,    42987672,      6561;\n  9: 19683, 387223659, 34570040220, 427278012876, 1366261865802, 1366261865802, 427278012876, 34570040220, 387223659, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(6*n-3)*x)*P(n-1,x) + (6*x-6*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 + (6*n - 3)*x)*P[n - 1, x] + (6*x - 6*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 + (6*n - 3)*x)*v[i - 1] + (6*x - 6*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":"A257625","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,        54,           9;\n  3:    27,       621,         621,           27;\n  4:    81,      6156,       18630,         6156,            81;\n  5:   243,     57591,      408726,       408726,         57591,           243;\n  6:   729,    526338,     7685847,     17166492,       7685847,        526338,          729;\n  7:  2187,   4765473,   132656859,    568014201,     568014201,     132656859,      4765473,        2187;\n  8:  6561,  42987672,  2175706332,  16305974568,   30672766854,   16305974568,   2175706332,    42987672,      6561;\n  9: 19683, 387223659, 34570040220, 427278012876, 1366261865802, 1366261865802, 427278012876, 34570040220, 387223659, 19683;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+(6*n-3)*x)*P(n-1,x) + (6*x-6*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 + (6*n - 3)*x)*P[n - 1, x] + (6*x - 6*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 + (6*n - 3)*x)*v[i - 1] + (6*x - 6*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-444025a5f2f6e0e5","steps":[],"edges":[],"text":"A257625"},"headline":"A257625: 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) = 6*n + 3.","chain":null}