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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":"7cbcb4168921fd27273d5544e91a3864aa2676aa3c0dbc3197e6ea36b54a706c","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-37ec1285a176250d","role":"sequence","name":"veronese_section_2_0 of Riordan array (1-x,x/(1-x)).","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,  1;\n  1: 1,  1;\n  2: 1,  3,   1;\n  3: 1,  7,   3;\n  4: 1, 13,   9,   1;\n  5: 1, 21,  25,   5;\n  6: 1, 31,  60,  19,   1;\n  7: 1, 43, 126,  63,   7;\n  8: 1, 57, 238, 182,  33, 1;\n  9: 1, 73, 414, 462, 129, 9;\n","recurrence_sentence":"The row polynomials satisfy -(6*n-6-(n^3-7*n+6)*x-(4*n^2-10*n+6)*x^2-(4*n-4)*x^3)*P(n,x) = -(6*n-6-(n^3-19*n+30)*x-(n^3+6*n^2-29*n+6)*x^2-(4*n^2+6*n-14)*x^3-(4*n+4)*x^4)*P(n-1,x) - (12*x-(2*n^2+6*n-4)*x^2+(2*n^2-2*n-20)*x^3+(8*n-4)*x^4+8*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + x, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1 + x;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 6 - (n^3 - 19*n + 30)*x - (n^3 + 6*n^2 - 29*n + 6)*x^2 - (4*n^2 + 6*n - 14)*x^3 - (4*n + 4)*x^4)*P[n - 1, x] - (12*x - (2*n^2 + 6*n - 4)*x^2 + (2*n^2 - 2*n - 20)*x^3 + (8*n - 4)*x^4 + 8*x^5)*D[P[n - 1, x], x])/(-6*n + 6 + (n^3 - 7*n + 6)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + x; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n - 6 - (n^3 - 19*n + 30)*x - (n^3 + 6*n^2 - 29*n + 6)*x^2 - (4*n^2 + 6*n - 14)*x^3 - (4*n + 4)*x^4)*v[i - 1] - (12*x - (2*n^2 + 6*n - 4)*x^2 + (2*n^2 - 2*n - 20)*x^3 + (8*n - 4)*x^4 + 8*x^5)*deriv(v[i - 1], x))/(-6*n + 6 + (n^3 - 7*n + 6)*x + (4*n^2 - 10*n + 6)*x^2 + (4*n - 4)*x^3)); 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-7717563017cb9e5f","steps":["Veronese 2","D+1"],"edges":["e-7643b829dc52d64b","e-fe25974f8b3acb67"],"text":"A159854 \u2192 Veronese 2 \u2192 D+1"},"headline":"A159854 \u2192 Veronese 2 \u2192 D+1","chain":null}