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Expand[Together[((n^2 + 10*n + 25 + (n^2 + 21*n + 87)*x - (2*n^2 - 112)*x^2 - (2*n^2 + 26*n - 65)*x^3 + (n^2 - 18*n + 17)*x^4 + (n^2 - 3*n + 2)*x^5)*P[n - 1, x] + ((2*n + 11)*x + (6*n + 48)*x^2 + (4*n + 82)*x^3 - (4*n - 68)*x^4 - (6*n - 27)*x^5 - (2*n - 4)*x^6)*D[P[n - 1, x], x] + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*D[P[n - 1, x], {x, 2}])/(1 + 2*x + x^2)]];\nTable[CoefficientList[P[n, x], 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^2 + 10*n + 25 + (n^2 + 21*n + 87)*x - (2*n^2 - 112)*x^2 - (2*n^2 + 26*n - 65)*x^3 + (n^2 - 18*n + 17)*x^4 + (n^2 - 3*n + 2)*x^5)*v[i - 1] + ((2*n + 11)*x + (6*n + 48)*x^2 + (4*n + 82)*x^3 - (4*n - 68)*x^4 - (6*n - 27)*x^5 - (2*n - 4)*x^6)*deriv(v[i - 1], x) + (x^2 + 5*x^3 + 10*x^4 + 10*x^5 + 5*x^6 + x^7)*deriv(deriv(v[i - 1], x), x))/(1 + 2*x + x^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-b8808e9caf8dcf35","steps":["x \u21a6 x/(x+1)"],"edges":["e-9418e8e8a896877a"],"text":"A371259 \u2192 x \u21a6 x/(x+1)"},"headline":"A371259 \u2192 x \u21a6 x/(x+1)","chain":null}