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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":"f241ce5fe85c077b1ebdabe17e5df966bd5773983c52c35858863ba5cdd1c38e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-948c4fae35fcbf15","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    1;\n   3:    2,     1;\n   4:   16,    15;\n   5:   71,    72,     1;\n   6:  211,   300,    45;\n   7:  496,  1485,   498,     1;\n   8: 1002,  7007,  3276,    91;\n   9: 1821, 27560, 18330,  1824,   1;\n  10: 3061, 90576, 99450, 19176, 153;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (127-189*x-3*x^2+65*x^3)*P(n-6,x) - (64-192*x+192*x^2-64*x^3)*d/dx P(n-6,x) + (3214-988*x-1186*x^2-16*x^3)*P(n-5,x) - (128-1264*x+1120*x^2+16*x^3)*d/dx P(n-5,x) + (10193+5330*x+61*x^2)*P(n-4,x) + (1152+5184*x+64*x^2)*d/dx P(n-4,x) - (188-468*x-32*x^2)*P(n-3,x) - (2048-416*x-32*x^2)*d/dx P(n-3,x) + (1361+131*x)*P(n-2,x) + (1472+128*x)*d/dx P(n-2,x) - (370+16*x)*P(n-1,x) - (384+16*x)*d/dx P(n-1,x) for n >= 7, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 2 + x, P(4,x) = 16 + 15*x, P(5,x) = 71 + 72*x + x^2, P(6,x) = 211 + 300*x + 45*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 2 + x;\nP[4, x_] := 16 + 15*x;\nP[5, x_] := 71 + 72*x + x^2;\nP[6, x_] := 211 + 300*x + 45*x^2;\nP[n_, x_] := P[n, x] = Expand[(127 - 189*x - 3*x^2 + 65*x^3)*P[n - 6, x] - (64 - 192*x + 192*x^2 - 64*x^3)*D[P[n - 6, x], x] + (3214 - 988*x - 1186*x^2 - 16*x^3)*P[n - 5, x] - (128 - 1264*x + 1120*x^2 + 16*x^3)*D[P[n - 5, x], x] + (10193 + 5330*x + 61*x^2)*P[n - 4, x] + (1152 + 5184*x + 64*x^2)*D[P[n - 4, x], x] - (188 - 468*x - 32*x^2)*P[n - 3, x] - (2048 - 416*x - 32*x^2)*D[P[n - 3, x], x] + (1361 + 131*x)*P[n - 2, x] + (1472 + 128*x)*D[P[n - 2, x], x] - (370 + 16*x)*P[n - 1, x] - (384 + 16*x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 2 + x; v[4] = 16 + 15*x; v[5] = 71 + 72*x + x^2; v[6] = 211 + 300*x + 45*x^2; for(n = 7, N, my(i = n); v[i] = (127 - 189*x - 3*x^2 + 65*x^3)*v[i - 6] - (64 - 192*x + 192*x^2 - 64*x^3)*deriv(v[i - 6], x) + (3214 - 988*x - 1186*x^2 - 16*x^3)*v[i - 5] - (128 - 1264*x + 1120*x^2 + 16*x^3)*deriv(v[i - 5], x) + (10193 + 5330*x + 61*x^2)*v[i - 4] + (1152 + 5184*x + 64*x^2)*deriv(v[i - 4], x) - (188 - 468*x - 32*x^2)*v[i - 3] - (2048 - 416*x - 32*x^2)*deriv(v[i - 3], x) + (1361 + 131*x)*v[i - 2] + (1472 + 128*x)*deriv(v[i - 2], x) - (370 + 16*x)*v[i - 1] - (384 + 16*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["Veronese 2","D+1"],"edges":["e-d4161fce9e8d4a19","e-1b5c998f0b493683"],"text":"A086645 \u2192 Veronese 2 \u2192 D+1"},"headline":"A086645 \u2192 Veronese 2 \u2192 D+1","chain":null}