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 1;\n  3:       15,       10,        1;\n  4:      105,      105,       25,        1;\n  5:      945,     1260,      490,       56,        1;\n  6:    10395,    17325,     9450,     1918,      119,       1;\n  7:   135135,   270270,   190575,    56980,     6825,     246,     1;\n  8:  2027025,  4729725,  4099095,  1636635,   302995,   22935,   501,    1;\n  9: 34459425, 91891800, 94594500, 47507460, 12122110, 1487200, 74316, 1012, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = n*x*P(n-1,x) + (x+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[n*x*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];\nTable[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], 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] = n*x*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":false}},{"oeis_id":"A269939","target":"triangle","first_row":1,"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: 0, 1;\n  2: 0, 1,    3;\n  3: 0, 1,   10,    15;\n  4: 0, 1,   25,   105,     105;\n  5: 0, 1,   56,   490,    1260,      945;\n  6: 0, 1,  119,  1918,    9450,    17325,    10395;\n  7: 0, 1,  246,  6825,   56980,   190575,   270270,   135135;\n  8: 0, 1,  501, 22935,  302995,  1636635,  4099095,  4729725,  2027025;\n  9: 0, 1, 1012, 74316, 1487200, 12122110, 47507460, 94594500, 91891800, 34459425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = n*x*P(n-1,x) + (x+x^2)*d/dx 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