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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":"5c68651455becab32c25c3a6a782a538c3e30a471477a65f12bd8f44407c3cf9","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-e5da0d00f07fcbc3","role":"sequence","name":"shift_x_plus_one of A133080 * A007318.","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,     1;\n  2:    8,     6,     1;\n  3:   28,    30,    10,     1;\n  4:   48,    80,    48,    12,     1;\n  5:  160,   320,   248,    92,    16,    1;\n  6:  256,   672,   720,   400,   120,   18,    1;\n  7:  832,  2464,  3072,  2080,   820,  186,   22,   1;\n  8: 1280,  4608,  7168,  6272,  3360, 1120,  224,  24,  1;\n  9: 4096, 16128, 27904, 27776, 17472, 7168, 1904, 312, 28, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n-2+2*n*x+x^2)*P(n,x) = -(5*n^3-4*n^2-11*n+10+(2*n^3+9*n^2-17*n-2)*x+(4*n^2+n-11)*x^2+(2*n-3)*x^3)*P(n-1,x) + (10*n^2+18*n+4+(9*n^2+37*n+22)*x+(2*n^2+22*n+28)*x^2+(4*n+13)*x^3+2*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[(-(5*n^3 - 4*n^2 - 11*n + 10 + (2*n^3 + 9*n^2 - 17*n - 2)*x + (4*n^2 + n - 11)*x^2 + (2*n - 3)*x^3)*P[n - 1, x] + (10*n^2 + 18*n + 4 + (9*n^2 + 37*n + 22)*x + (2*n^2 + 22*n + 28)*x^2 + (4*n + 13)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^2 + n - 2 + 2*n*x + 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] = (-(5*n^3 - 4*n^2 - 11*n + 10 + (2*n^3 + 9*n^2 - 17*n - 2)*x + (4*n^2 + n - 11)*x^2 + (2*n - 3)*x^3)*v[i - 1] + (10*n^2 + 18*n + 4 + (9*n^2 + 37*n + 22)*x + (2*n^2 + 22*n + 28)*x^2 + (4*n + 13)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^2 + n - 2 + 2*n*x + 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-d75669a8cd7f9393","steps":["x \u21a6 x+1","D+1"],"edges":["e-52dd73548014190c","e-e7cb09aac56cc482"],"text":"A133087 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A133087 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}