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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":"9f7d865c41a6275c84b91e3a457cd04d8f6b0091055836e7e5c036a35aed8169","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:  1;\n  1:  4,   2;\n  2:  5,  12,    4;\n  3: 12,  42,   40,     8;\n  4:  9,  56,  120,    96,    16;\n  5: 20, 146,  400,   496,   256,    32;\n  6: 13, 132,  540,  1120,  1200,   576,    64;\n  7: 28, 314, 1464,  3640,  5120,  3936,  1408,   128;\n  8: 17, 240, 1456,  4928, 10080, 12544,  8960,  3072,  256;\n  9: 36, 546, 3616, 13664, 32256, 48832, 46592, 26112, 7168, 512;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-16*n+2+(16*n-24)*x+8*x^2)*P(n,x) = -(24*n^3-96*n^2+102*n-6+(32*n^3-64*n^2-48*n+124)*x+(64*n^2-136*n+56)*x^2+(32*n-48)*x^3)*P(n-1,x) + (12*n^2-16*n+1+(40*n^2-24*n-18)*x+(32*n^2+48*n-44)*x^2+(64*n+8)*x^3+32*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(24*n^3 - 96*n^2 + 102*n - 6 + (32*n^3 - 64*n^2 - 48*n + 124)*x + (64*n^2 - 136*n + 56)*x^2 + (32*n - 48)*x^3)*P[n - 1, x] + (12*n^2 - 16*n + 1 + (40*n^2 - 24*n - 18)*x + (32*n^2 + 48*n - 44)*x^2 + (64*n + 8)*x^3 + 32*x^4)*D[P[n - 1, x], x])/(8*n^2 - 16*n + 2 + (16*n - 24)*x + 8*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] = (-(24*n^3 - 96*n^2 + 102*n - 6 + (32*n^3 - 64*n^2 - 48*n + 124)*x + (64*n^2 - 136*n + 56)*x^2 + (32*n - 48)*x^3)*v[i - 1] + (12*n^2 - 16*n + 1 + (40*n^2 - 24*n - 18)*x + (32*n^2 + 48*n - 44)*x^2 + (64*n + 8)*x^3 + 32*x^4)*deriv(v[i - 1], x))/(8*n^2 - 16*n + 2 + (16*n - 24)*x + 8*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-ea2139b8dbf6bf8d","steps":["basis A013609","D+1"],"edges":["e-d9920fdddfeeaf81","e-3f6c5f175eb2dbc6"],"text":"A133080 \u2192 basis A013609 \u2192 D+1"},"headline":"A133080 \u2192 basis A013609 \u2192 D+1","chain":null}