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(1+(3*n+3)*x+(3*n^2+9*n+2)*x^2+(n^3+9*n^2+8*n-2)*x^3+(3*n^3+9*n^2-3)*x^4+(3*n^3+3*n^2-3*n-1)*x^5+(n^3-n)*x^6)*P(n-3,x) - (3+(9*n+3)*x+(9*n^2+12*n-9)*x^2+(3*n^3+15*n^2-9*n-15)*x^3+(6*n^3+3*n^2-18*n-6)*x^4+(3*n^3-3*n^2-6*n)*x^5)*P(n-2,x) + (3+(9*n-3)*x+(9*n^2-3*n-9)*x^2+(3*n^3+3*n^2-15*n+3)*x^3+(3*n^3-6*n^2-3*n+6)*x^4)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1 + 4*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1 + 4*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((1 + (3*n + 3)*x + (3*n^2 + 9*n + 2)*x^2 + (n^3 + 9*n^2 + 8*n - 2)*x^3 + (3*n^3 + 9*n^2 - 3)*x^4 + (3*n^3 + 3*n^2 - 3*n - 1)*x^5 + (n^3 - n)*x^6)*P[n - 3, x] - (3 + (9*n + 3)*x + (9*n^2 + 12*n - 9)*x^2 + (3*n^3 + 15*n^2 - 9*n - 15)*x^3 + (6*n^3 + 3*n^2 - 18*n - 6)*x^4 + (3*n^3 - 3*n^2 - 6*n)*x^5)*P[n - 2, x] + (3 + (9*n - 3)*x + (9*n^2 - 3*n - 9)*x^2 + (3*n^3 + 3*n^2 - 15*n + 3)*x^3 + (3*n^3 - 6*n^2 - 3*n + 6)*x^4)*P[n - 1, x])/(1 + (3*n - 3)*x + (3*n^2 - 6*n + 2)*x^2 + (n^3 - 3*n^2 + 2*n)*x^3)]];\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] = 1 + 4*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((1 + (3*n + 3)*x + (3*n^2 + 9*n + 2)*x^2 + (n^3 + 9*n^2 + 8*n - 2)*x^3 + (3*n^3 + 9*n^2 - 3)*x^4 + (3*n^3 + 3*n^2 - 3*n - 1)*x^5 + (n^3 - n)*x^6)*v[i - 3] - (3 + (9*n + 3)*x + (9*n^2 + 12*n - 9)*x^2 + (3*n^3 + 15*n^2 - 9*n - 15)*x^3 + (6*n^3 + 3*n^2 - 18*n - 6)*x^4 + (3*n^3 - 3*n^2 - 6*n)*x^5)*v[i - 2] + (3 + (9*n - 3)*x + (9*n^2 - 3*n - 9)*x^2 + (3*n^3 + 3*n^2 - 15*n + 3)*x^3 + (3*n^3 - 6*n^2 - 3*n + 6)*x^4)*v[i - 1])/(1 + (3*n - 3)*x + (3*n^2 - 6*n + 2)*x^2 + (n^3 - 3*n^2 + 2*n)*x^3)); 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":["\u00d7(\u2212(1/2)n+(1/2)n\u00b2)","\u03b8+1"],"edges":["e-ccfad80d3bf960c1","e-21ef48566d5646a5"],"text":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 \u03b8+1"},"headline":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 \u03b8+1","chain":null}