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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":"d1bf0d34480610e915c063a484460a50ae7fde633f1ede521dc0ac69cff23384","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-6feee566fc77ea06","role":"sequence","name":"veronese_section_2_1 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.","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:   4;\n  1:   9,    1;\n  2:  18,    6;\n  3:  35,   21,    1;\n  4:  64,   60,    8;\n  5: 109,  153,   37,   1;\n  6: 174,  354,  134,  10;\n  7: 263,  749,  418,  57,  1;\n  8: 380, 1464, 1160, 248, 12;\n  9: 529, 2673, 2910, 910, 81, 1;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-48*n^2+152*n-24-(8*n^3-68*n^2+294*n+30)*x+(n^3-30*n^2+123*n+162)*x^2+(4*n^2-30*n-78)*x^3+(4*n+12)*x^4)*P(n,x) = (16*n^3+104*n+96+(8*n^3-28*n^2-142*n-408)*x-(7*n^3-59*n^2+210*n-668)*x^2+(n^3-29*n^2+134*n-276)*x^3+(4*n^2-34*n+40)*x^4+4*n*x^5)*P(n-1,x) + ((32*n^2-32*n+80)*x-(48*n^2-160*n+372)*x^2+(18*n^2-190*n+504)*x^3-(2*n^2-70*n+280)*x^4-(8*n-76)*x^5-8*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4.","mathematica":"P[0, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 + 104*n + 96 + (8*n^3 - 28*n^2 - 142*n - 408)*x - (7*n^3 - 59*n^2 + 210*n - 668)*x^2 + (n^3 - 29*n^2 + 134*n - 276)*x^3 + (4*n^2 - 34*n + 40)*x^4 + 4*n*x^5)*P[n - 1, x] + ((32*n^2 - 32*n + 80)*x - (48*n^2 - 160*n + 372)*x^2 + (18*n^2 - 190*n + 504)*x^3 - (2*n^2 - 70*n + 280)*x^4 - (8*n - 76)*x^5 - 8*x^6)*D[P[n - 1, x], x])/(16*n^3 - 48*n^2 + 152*n - 24 - (8*n^3 - 68*n^2 + 294*n + 30)*x + (n^3 - 30*n^2 + 123*n + 162)*x^2 + (4*n^2 - 30*n - 78)*x^3 + (4*n + 12)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 + 104*n + 96 + (8*n^3 - 28*n^2 - 142*n - 408)*x - (7*n^3 - 59*n^2 + 210*n - 668)*x^2 + (n^3 - 29*n^2 + 134*n - 276)*x^3 + (4*n^2 - 34*n + 40)*x^4 + 4*n*x^5)*v[i - 1] + ((32*n^2 - 32*n + 80)*x - (48*n^2 - 160*n + 372)*x^2 + (18*n^2 - 190*n + 504)*x^3 - (2*n^2 - 70*n + 280)*x^4 - (8*n - 76)*x^5 - 8*x^6)*deriv(v[i - 1], x))/(16*n^3 - 48*n^2 + 152*n - 24 - (8*n^3 - 68*n^2 + 294*n + 30)*x + (n^3 - 30*n^2 + 123*n + 162)*x^2 + (4*n^2 - 30*n - 78)*x^3 + (4*n + 12)*x^4)); 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-f48bed27dbc29c5f","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-a61d9f5d8a4277bd","e-ed7f9346713295d3"],"text":"A391533 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A391533 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}