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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":"ba0a13571388238206895c2315e4fc949cde8d3c4c40ea69675163e45f566caa","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-cbf81d8f5ae9d341","role":"sequence","name":"veronese_section_2_0 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:   5,    1;\n  1:   9,    5;\n  2:  17,   15,    1;\n  3:  29,   39,    7;\n  4:  45,   93,   29,    1;\n  5:  65,  201,   97,    9;\n  6:  89,  395,  284,   47,   1;\n  7: 117,  715,  742,  191,  11;\n  8: 149, 1209, 1750,  662,  69,  1;\n  9: 185, 1933, 3774, 2022, 329, 13;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-16*n^2+40*n-(8*n^3-52*n^2+210*n-30)*x+(n^3-28*n^2+93*n+102)*x^2+(4*n^2-26*n-66)*x^3+(4*n+12)*x^4)*P(n,x) = (16*n^3+16*n^2+40*n+(8*n^3+12*n^2-314*n+78)*x-(7*n^3-36*n^2+23*n-206)*x^2+(n^3-26*n^2+67*n-98)*x^3+(4*n^2-26*n+2)*x^4+(4*n+4)*x^5)*P(n-1,x) + ((32*n^2+32*n-80)*x-(48*n^2-64*n+28)*x^2+(18*n^2-154*n+252)*x^3-(2*n^2-66*n+204)*x^4-(8*n-68)*x^5-8*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 5 + x.","mathematica":"P[0, x_] := 5 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 + 16*n^2 + 40*n + (8*n^3 + 12*n^2 - 314*n + 78)*x - (7*n^3 - 36*n^2 + 23*n - 206)*x^2 + (n^3 - 26*n^2 + 67*n - 98)*x^3 + (4*n^2 - 26*n + 2)*x^4 + (4*n + 4)*x^5)*P[n - 1, x] + ((32*n^2 + 32*n - 80)*x - (48*n^2 - 64*n + 28)*x^2 + (18*n^2 - 154*n + 252)*x^3 - (2*n^2 - 66*n + 204)*x^4 - (8*n - 68)*x^5 - 8*x^6)*D[P[n - 1, x], x])/(16*n^3 - 16*n^2 + 40*n - (8*n^3 - 52*n^2 + 210*n - 30)*x + (n^3 - 28*n^2 + 93*n + 102)*x^2 + (4*n^2 - 26*n - 66)*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] = 5 + x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 + 16*n^2 + 40*n + (8*n^3 + 12*n^2 - 314*n + 78)*x - (7*n^3 - 36*n^2 + 23*n - 206)*x^2 + (n^3 - 26*n^2 + 67*n - 98)*x^3 + (4*n^2 - 26*n + 2)*x^4 + (4*n + 4)*x^5)*v[i - 1] + ((32*n^2 + 32*n - 80)*x - (48*n^2 - 64*n + 28)*x^2 + (18*n^2 - 154*n + 252)*x^3 - (2*n^2 - 66*n + 204)*x^4 - (8*n - 68)*x^5 - 8*x^6)*deriv(v[i - 1], x))/(16*n^3 - 16*n^2 + 40*n - (8*n^3 - 52*n^2 + 210*n - 30)*x + (n^3 - 28*n^2 + 93*n + 102)*x^2 + (4*n^2 - 26*n - 66)*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","D+1"],"edges":["e-9f9d452726caee29","e-33455e91eedb8ab8"],"text":"A391533 \u2192 Veronese 2 \u2192 D+1"},"headline":"A391533 \u2192 Veronese 2 \u2192 D+1","chain":null}