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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":"09593a59af7375f750ce1d5f59c13dfceabb41d929f59cc39bd3dd87188e99c7","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-1f922d5ae3a4fc22","role":"sequence","name":"veronese_section_2_1 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: 0;\n  1: 1,   1;\n  2: 2,   2;\n  3: 3,   5,   1;\n  4: 4,  12,   4;\n  5: 5,  25,  13,   1;\n  6: 6,  46,  38,   6;\n  7: 7,  77,  98,  25,  1;\n  8: 8, 120, 224,  88,  8;\n  9: 9, 177, 462, 270, 41, 1;\n","recurrence_sentence":"The row polynomials satisfy -(2*n-2-(n^3-2*n^2-n+2)*x-(4*n^2-14*n+10)*x^2-(4*n-4)*x^3)*P(n,x) = -(2*n-(n^3-n^2-6*n+12)*x-(n^3+3*n^2-26*n+20)*x^2-(4*n^2-2*n-8)*x^3-4*n*x^4)*P(n-1,x) - (4*x-(2*n^2+2*n-8)*x^2+(2*n^2-6*n-8)*x^3+(8*n-12)*x^4+8*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - (n^3 - n^2 - 6*n + 12)*x - (n^3 + 3*n^2 - 26*n + 20)*x^2 - (4*n^2 - 2*n - 8)*x^3 - 4*n*x^4)*P[n - 1, x] - (4*x - (2*n^2 + 2*n - 8)*x^2 + (2*n^2 - 6*n - 8)*x^3 + (8*n - 12)*x^4 + 8*x^5)*D[P[n - 1, x], x])/(-2*n + 2 + (n^3 - 2*n^2 - n + 2)*x + (4*n^2 - 14*n + 10)*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] = 0; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - (n^3 - n^2 - 6*n + 12)*x - (n^3 + 3*n^2 - 26*n + 20)*x^2 - (4*n^2 - 2*n - 8)*x^3 - 4*n*x^4)*v[i - 1] - (4*x - (2*n^2 + 2*n - 8)*x^2 + (2*n^2 - 6*n - 8)*x^3 + (8*n - 12)*x^4 + 8*x^5)*deriv(v[i - 1], x))/(-2*n + 2 + (n^3 - 2*n^2 - n + 2)*x + (4*n^2 - 14*n + 10)*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 (j=1)","D+1"],"edges":["e-6fdf1329e5e61ed8","e-b07c08e411543e96"],"text":"A159854 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A159854 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}