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begins:\n  0: 0;\n  1: 1;\n  2: 2,   8;\n  3: 3,  24,   27;\n  4: 4,  48,  108,   64;\n  5: 5,  80,  270,  320,   125;\n  6: 6, 120,  540,  960,   750,   216;\n  7: 7, 168,  945, 2240,  2625,  1512,   343;\n  8: 8, 224, 1512, 4480,  7000,  6048,  2744,  512;\n  9: 9, 288, 2268, 8064, 15750, 18144, 12348, 4608, 729;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n+(4*n^2-4*n)*x-(9*n^2-9*n)*x^2)*P(n-2,x) + ((4*n^2-4*n)*x^2+(4*n^2-4*n)*x^3)*d/dx P(n-2,x) + (2*n^2-4*n+(4*n^2-8*n)*x)*P(n-1,x) - (4*n^2-8*n)*x^2*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 2 + 8*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n + (4*n^2 - 4*n)*x - (9*n^2 - 9*n)*x^2)*P[n - 2, x] + ((4*n^2 - 4*n)*x^2 + (4*n^2 - 4*n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 4*n + (4*n^2 - 8*n)*x)*P[n - 1, x] - (4*n^2 - 8*n)*x^2*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] = 1; v[3] = 2 + 8*x; for(n = 3, N, my(i = n + 1); v[i] = (-(n^2 - n + (4*n^2 - 4*n)*x - (9*n^2 - 9*n)*x^2)*v[i - 2] + ((4*n^2 - 4*n)*x^2 + (4*n^2 - 4*n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 4*n + (4*n^2 - 8*n)*x)*v[i - 1] - (4*n^2 - 8*n)*x^2*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-c8c19eba9d68881f","steps":["D","(\u03b8+1)\u00b2"],"edges":["e-fd16b43297f2bb3e","e-1e3b90e912ae07d0"],"text":"A007318 \u2192 D \u2192 (\u03b8+1)\u00b2"},"headline":"A007318 \u2192 D \u2192 (\u03b8+1)\u00b2","chain":null}