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(6*n^2+2*n-8-20*x-(3*n^2+4*n-35)*x^2+4*x^3-16*x^4)*d/dx P(n-2,x) + (8*x^2-16*x^3+8*x^4)*d^2/dx^2 P(n-2,x) + (6*n+26+(16*n-28)*x+16*x^2)*P(n-1,x) for n >= 5, with P(1,x) = 8 + 9*x + x^2, P(2,x) = 22 + 63*x + 24*x^2 + x^3, P(3,x) = 52 + 373*x + 314*x^2 + 55*x^3 + x^4, P(4,x) = 114 + 1739*x + 3252*x^2 + 1265*x^3 + 118*x^4 + x^5.","mathematica":"P[1, x_] := 8 + 9*x + x^2;\nP[2, x_] := 22 + 63*x + 24*x^2 + x^3;\nP[3, x_] := 52 + 373*x + 314*x^2 + 55*x^3 + x^4;\nP[4, x_] := 114 + 1739*x + 3252*x^2 + 1265*x^3 + 118*x^4 + x^5;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n^3 + 4*n^2 - 16*n + (6*n^3 - 10*n^2 - 20*n)*x - (14*n^2 + 14*n)*x^2 - (3*n^3 - 2*n^2 - 9*n)*x^3)*P[n - 4, x] + ((6*n^3 + 2*n^2 - 8*n)*x - (6*n^2 + 6*n)*x^2 - (3*n^3 - 2*n^2 - 9*n)*x^3)*D[P[n - 4, x], x] + (12*n^3 - 14*n^2 - 22*n + 24 + (6*n^3 - 16*n^2 - 16*n + 6)*x - (25*n^2 - 6*n - 35)*x^2 - (3*n^3 - 5*n^2 + 9*n + 25)*x^3)*P[n - 3, x] + (18*n^2 + 6*n - 24 + (6*n^3 - 22*n^2 - 34*n + 14)*x - (15*n^2 - 18*n - 41)*x^2 - (3*n^3 - 8*n^2 - 11*n + 12)*x^3 + (3*n^2 - 2*n - 9)*x^4)*D[P[n - 3, x], x] + ((6*n^2 + 2*n - 8)*x - (6*n^2 + 8*n - 2)*x^2 - (3*n^2 - 8*n - 15)*x^3 + (3*n^2 - 2*n - 9)*x^4)*D[P[n - 3, x], {x, 2}] - (6*n^2 + 14*n + 12 + (12*n^2 - 36*n - 84)*x + (5*n^2 - 10*n + 25)*x^2 + 16*n*x^3)*P[n - 2, x] - (6*n^2 + 2*n - 8 - 20*x - (3*n^2 + 4*n - 35)*x^2 + 4*x^3 - 16*x^4)*D[P[n - 2, x], x] + (8*x^2 - 16*x^3 + 8*x^4)*D[P[n - 2, x], {x, 2}] + (6*n + 26 + (16*n - 28)*x + 16*x^2)*P[n - 1, x])/(8)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 8 + 9*x + x^2; v[2] = 22 + 63*x + 24*x^2 + x^3; v[3] = 52 + 373*x + 314*x^2 + 55*x^3 + x^4; v[4] = 114 + 1739*x + 3252*x^2 + 1265*x^3 + 118*x^4 + x^5; for(n = 5, N, my(i = n); v[i] = ((12*n^3 + 4*n^2 - 16*n + (6*n^3 - 10*n^2 - 20*n)*x - (14*n^2 + 14*n)*x^2 - (3*n^3 - 2*n^2 - 9*n)*x^3)*v[i - 4] + ((6*n^3 + 2*n^2 - 8*n)*x - (6*n^2 + 6*n)*x^2 - (3*n^3 - 2*n^2 - 9*n)*x^3)*deriv(v[i - 4], x) + (12*n^3 - 14*n^2 - 22*n + 24 + (6*n^3 - 16*n^2 - 16*n + 6)*x - (25*n^2 - 6*n - 35)*x^2 - (3*n^3 - 5*n^2 + 9*n + 25)*x^3)*v[i - 3] + (18*n^2 + 6*n - 24 + (6*n^3 - 22*n^2 - 34*n + 14)*x - (15*n^2 - 18*n - 41)*x^2 - (3*n^3 - 8*n^2 - 11*n + 12)*x^3 + (3*n^2 - 2*n - 9)*x^4)*deriv(v[i - 3], x) + ((6*n^2 + 2*n - 8)*x - (6*n^2 + 8*n - 2)*x^2 - (3*n^2 - 8*n - 15)*x^3 + (3*n^2 - 2*n - 9)*x^4)*deriv(deriv(v[i - 3], x), x) - (6*n^2 + 14*n + 12 + (12*n^2 - 36*n - 84)*x + (5*n^2 - 10*n + 25)*x^2 + 16*n*x^3)*v[i - 2] - (6*n^2 + 2*n - 8 - 20*x - (3*n^2 + 4*n - 35)*x^2 + 4*x^3 - 16*x^4)*deriv(v[i - 2], x) + (8*x^2 - 16*x^3 + 8*x^4)*deriv(deriv(v[i - 2], x), x) + (6*n + 26 + (16*n - 28)*x + 16*x^2)*v[i - 1])/(8)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-09153823987c7e65","steps":["D+1"],"edges":["e-a8b72947c1839ff2"],"text":"A046739 \u2192 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