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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":"6c84293a881fe6a353fa2056698cb909a04f61a1a696fb94066b0c3727ba3905","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-33a50ee3b762dcdc","role":"sequence","name":"basis_map_A008288 of Triangle of coefficients in expansion of (1+2*x)^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:      2;\n  2:     16,       8;\n  3:     82,     104,      24;\n  4:    352,     784,     432,      64;\n  5:   1378,    4560,    4320,    1472,     160;\n  6:   5104,   22808,   32016,   18368,    4480,     384;\n  7:  18226,  103480,  197208,  165952,   67040,   12672,    896;\n  8:  63424,  438816, 1071648, 1219968,  715520,  221184,  34048,  2048;\n  9: 216514, 1771424, 5321856, 7765632, 6125760, 2724864, 679168, 88064, 4608;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9-12*x+4*x^2)*P(n-4,x) + (24-4*x-8*x^2)*P(n-3,x) - (22+12*x+4*x^2)*P(n-2,x) + (8+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 16 + 8*x, P(3,x) = 82 + 104*x + 24*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 16 + 8*x;\nP[3, x_] := 82 + 104*x + 24*x^2;\nP[n_, x_] := P[n, x] = Expand[-(9 - 12*x + 4*x^2)*P[n - 4, x] + (24 - 4*x - 8*x^2)*P[n - 3, x] - (22 + 12*x + 4*x^2)*P[n - 2, x] + (8 + 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] = 2; v[3] = 16 + 8*x; v[4] = 82 + 104*x + 24*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(9 - 12*x + 4*x^2)*v[i - 4] + (24 - 4*x - 8*x^2)*v[i - 3] - (22 + 12*x + 4*x^2)*v[i - 2] + (8 + 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-71965e79e47a4526","steps":["basis A008288","D"],"edges":["e-9c90c90150c15b0e","e-56503b6e9835aeb1"],"text":"A013609 \u2192 basis A008288 \u2192 D"},"headline":"A013609 \u2192 basis A008288 \u2192 D","chain":null}