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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":"2891ad99e5354d989ea1fda9aba62102e6577b58142b2a8de0488c857e219642","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-f1bdd4180f3e6ae5","role":"sequence","name":"basis_map_A013609 of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","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:  2;\n  2:  4,   8;\n  3: 10,  32,   24;\n  4:  8,  48,   96,    64;\n  5: 18, 128,  336,   384,   160;\n  6: 12, 120,  480,   960,   960,   384;\n  7: 26, 288, 1320,  3200,  4320,  3072,   896;\n  8: 16, 224, 1344,  4480,  8960, 10752,  7168,  2048;\n  9: 34, 512, 3360, 12544, 29120, 43008, 39424, 20480, 4608;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-6*n+4)*P(n,x) = -(6*n^3-30*n^2+44*n-16+(8*n^3-36*n^2+44*n-8)*x)*P(n-1,x) + (3*n^2-7*n+3+(10*n^2-22*n+8)*x+(8*n^2-16*n+4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 4 + 8*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 4 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n^3 - 30*n^2 + 44*n - 16 + (8*n^3 - 36*n^2 + 44*n - 8)*x)*P[n - 1, x] + (3*n^2 - 7*n + 3 + (10*n^2 - 22*n + 8)*x + (8*n^2 - 16*n + 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 6*n + 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; v[3] = 4 + 8*x; for(n = 3, N, my(i = n + 1); v[i] = (-(6*n^3 - 30*n^2 + 44*n - 16 + (8*n^3 - 36*n^2 + 44*n - 8)*x)*v[i - 1] + (3*n^2 - 7*n + 3 + (10*n^2 - 22*n + 8)*x + (8*n^2 - 16*n + 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 6*n + 4)); 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-ea2139b8dbf6bf8d","steps":["basis A013609","D"],"edges":["e-d9920fdddfeeaf81","e-fd97c1554347ba4f"],"text":"A133080 \u2192 basis A013609 \u2192 D"},"headline":"A133080 \u2192 basis A013609 \u2192 D","chain":null}