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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"6e813bab04da718525e035417be7b03ea9dce41ffc90775aa9eb13d03cb2c950","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e7d0ffa461d9dbc4","role":"sequence","name":"Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  0;\n  1:  4;\n  2:  8,  18;\n  3: 12,  54,   48;\n  4: 16, 108,  192,   100;\n  5: 20, 180,  480,   500,   180;\n  6: 24, 270,  960,  1500,  1080,   294;\n  7: 28, 378, 1680,  3500,  3780,  2058,   448;\n  8: 32, 504, 2688,  7000, 10080,  8232,  3584,  648;\n  9: 36, 648, 4032, 12600, 22680, 24696, 16128, 5832, 900;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+4*x+6*x^2+4*x^3+x^4)*P(n-4,x) + (4+12*x+12*x^2+4*x^3)*P(n-3,x) - (6+12*x+6*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 4, P(2,x) = 8 + 18*x, P(3,x) = 12 + 54*x + 48*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[2, x_] := 8 + 18*x;\nP[3, x_] := 12 + 54*x + 48*x^2;\nP[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*P[n - 4, x] + (4 + 12*x + 12*x^2 + 4*x^3)*P[n - 3, x] - (6 + 12*x + 6*x^2)*P[n - 2, x] + (4 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; v[3] = 8 + 18*x; v[4] = 12 + 54*x + 48*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(1 + 4*x + 6*x^2 + 4*x^3 + x^4)*v[i - 4] + (4 + 12*x + 12*x^2 + 4*x^3)*v[i - 3] - (6 + 12*x + 6*x^2)*v[i - 2] + (4 + 4*x)*v[i - 1]); 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":["(\u03b8+1)\u00b2","D"],"edges":["e-6f35c326bbf0f62d","e-c8fd413bc255cf6f"],"text":"A007318 \u2192 (\u03b8+1)\u00b2 \u2192 D"},"headline":"A007318 \u2192 (\u03b8+1)\u00b2 \u2192 D","chain":null}