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4: 12,   36,    12;\n  5: 20,  140,   120,     20;\n  6: 30,  450,   750,    300,    30;\n  7: 42, 1302,  3780,   2730,   630,    42;\n  8: 56, 3528, 16856,  19600,  7840,  1176,   56;\n  9: 72, 9144, 69552, 122472, 75600, 19152, 2016, 72;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n+(3*n^2-3*n)*x+(n^2-n)*x^2)*P(n-2,x) - ((3*n^2-3*n)*x+(2*n^2-2*n)*x^2)*d/dx P(n-2,x) - (n^2-n)*x^2*d^2/dx^2 P(n-2,x) + (2*n^2-4*n+(2*n^2-4*n)*x)*P(n-1,x) + (2*n^2-4*n)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (3*n^2 - 3*n)*x + (n^2 - n)*x^2)*P[n - 2, x] - ((3*n^2 - 3*n)*x + (2*n^2 - 2*n)*x^2)*D[P[n - 2, x], x] - (n^2 - n)*x^2*D[P[n - 2, x], {x, 2}] + (2*n^2 - 4*n + (2*n^2 - 4*n)*x)*P[n - 1, x] + (2*n^2 - 4*n)*x*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = (-(n^2 - n + (3*n^2 - 3*n)*x + (n^2 - n)*x^2)*v[i - 2] - ((3*n^2 - 3*n)*x + (2*n^2 - 2*n)*x^2)*deriv(v[i - 2], x) - (n^2 - n)*x^2*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 4*n + (2*n^2 - 4*n)*x)*v[i - 1] + (2*n^2 - 4*n)*x*deriv(v[i - 1], x))/(n^2 - 3*n + 2)); 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}}],"derivation":{"root":"n-d6ec17c5486fb722","steps":["D","basis A008277"],"edges":["e-1aec373ef809ab0b","e-fb51abecbcdfb975"],"text":"A127648 \u2192 D \u2192 basis A008277"},"headline":"A127648 \u2192 D \u2192 basis A008277","chain":null}