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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":"ff9b17bbdb1df2c20b2f5c6d7ab05cf77db19ef58f0e70b881414e39d46e29e6","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-c498b183c57d9f9c","role":"sequence","name":"veronese_section_2_0 of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","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:   1;\n  2:   2,    1;\n  3:   7,    6;\n  4:  16,   17,    1;\n  5:  29,   58,   15;\n  6:  46,  185,   73,    1;\n  7:  67,  486,  294,   28;\n  8:  92, 1081, 1125,  214,  1;\n  9: 121, 2122, 3773, 1104, 45;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12*x^2+5*x^3)*P(n-6,x) + 4*x^3*d/dx P(n-6,x) - (24*x+27*x^2+4*x^3)*P(n-5,x) - (12*x^2+4*x^3)*d/dx P(n-5,x) + (12+15*x-3*x^2)*P(n-4,x) + 12*x*d/dx P(n-4,x) + (23+6*x+8*x^2)*P(n-3,x) - (4+4*x-8*x^2)*d/dx P(n-3,x) - (3-15*x)*P(n-2,x) + (8+12*x)*d/dx P(n-2,x) + (1-4*x)*P(n-1,x) - (4+4*x)*d/dx P(n-1,x) for n >= 6, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 2 + x, P(3,x) = 7 + 6*x, P(4,x) = 16 + 17*x + x^2, P(5,x) = 29 + 58*x + 15*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[3, x_] := 7 + 6*x;\nP[4, x_] := 16 + 17*x + x^2;\nP[5, x_] := 29 + 58*x + 15*x^2;\nP[n_, x_] := P[n, x] = Expand[(12*x^2 + 5*x^3)*P[n - 6, x] + 4*x^3*D[P[n - 6, x], x] - (24*x + 27*x^2 + 4*x^3)*P[n - 5, x] - (12*x^2 + 4*x^3)*D[P[n - 5, x], x] + (12 + 15*x - 3*x^2)*P[n - 4, x] + 12*x*D[P[n - 4, x], x] + (23 + 6*x + 8*x^2)*P[n - 3, x] - (4 + 4*x - 8*x^2)*D[P[n - 3, x], x] - (3 - 15*x)*P[n - 2, x] + (8 + 12*x)*D[P[n - 2, x], x] + (1 - 4*x)*P[n - 1, x] - (4 + 4*x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 2 + x; v[4] = 7 + 6*x; v[5] = 16 + 17*x + x^2; v[6] = 29 + 58*x + 15*x^2; for(n = 6, N, my(i = n + 1); v[i] = (12*x^2 + 5*x^3)*v[i - 6] + 4*x^3*deriv(v[i - 6], x) - (24*x + 27*x^2 + 4*x^3)*v[i - 5] - (12*x^2 + 4*x^3)*deriv(v[i - 5], x) + (12 + 15*x - 3*x^2)*v[i - 4] + 12*x*deriv(v[i - 4], x) + (23 + 6*x + 8*x^2)*v[i - 3] - (4 + 4*x - 8*x^2)*deriv(v[i - 3], x) - (3 - 15*x)*v[i - 2] + (8 + 12*x)*deriv(v[i - 2], x) + (1 - 4*x)*v[i - 1] - (4 + 4*x)*deriv(v[i - 1], x)); 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-6758fe7729845be8","steps":["Veronese 2","D+1"],"edges":["e-b36b946d9b6c256e","e-f1c845c41e494fca"],"text":"A054142 \u2192 Veronese 2 \u2192 D+1"},"headline":"A054142 \u2192 Veronese 2 \u2192 D+1","chain":null}