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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":"54027fa7a1386b75920f6634b9629dbc36d87fadfdc1845961f1b83e7534c29f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-dd70177049267f47","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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:     24,      15,       2;\n  1:     78,      78,      23,       2;\n  2:    244,     341,     162,      31,       2;\n  3:    743,    1344,     897,     276,      39,       2;\n  4:   2218,    4944,    4290,    1854,     420,      47,      2;\n  5:   6521,   17314,   18585,   10450,    3320,     594,     55,      2;\n  6:  18942,   58443,   74974,   52211,   21570,    5403,    798,     63,     2;\n  7:  54486,  191718,  286621,  238868,  122226,   39774,   8211,   1032,    71,    2;\n  8: 155462,  614737, 1050696, 1021858,  625982,  252189,  67564,  11852,  1296,   79,  2;\n  9: 440557, 1934760, 3724170, 4146312, 2965815, 1430324, 474110, 107820, 16434, 1590, 87, 2;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-26*n-42+(4*n^3+18*n^2+12*n)*x-(18*n+24)*x^2-(4*n+6)*x^3)*P(n,x) = (3*n^3+4*n^2-60*n-124+(7*n^3+42*n^2+34*n-12)*x+(2*n^3+14*n^2-15*n-46)*x^2-(17*n+30)*x^3-(2*n+4)*x^4)*P(n-1,x) + (5*n^2+5*n-70+(16*n^2+46*n-149)*x+(13*n^2+49*n-157)*x^2+(2*n^2+8*n-101)*x^3-25*x^4-2*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 24 + 15*x + 2*x^2.","mathematica":"P[0, x_] := 24 + 15*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^3 + 4*n^2 - 60*n - 124 + (7*n^3 + 42*n^2 + 34*n - 12)*x + (2*n^3 + 14*n^2 - 15*n - 46)*x^2 - (17*n + 30)*x^3 - (2*n + 4)*x^4)*P[n - 1, x] + (5*n^2 + 5*n - 70 + (16*n^2 + 46*n - 149)*x + (13*n^2 + 49*n - 157)*x^2 + (2*n^2 + 8*n - 101)*x^3 - 25*x^4 - 2*x^5)*D[P[n - 1, x], x])/(2*n^3 - 26*n - 42 + (4*n^3 + 18*n^2 + 12*n)*x - (18*n + 24)*x^2 - (4*n + 6)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 24 + 15*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^3 + 4*n^2 - 60*n - 124 + (7*n^3 + 42*n^2 + 34*n - 12)*x + (2*n^3 + 14*n^2 - 15*n - 46)*x^2 - (17*n + 30)*x^3 - (2*n + 4)*x^4)*v[i - 1] + (5*n^2 + 5*n - 70 + (16*n^2 + 46*n - 149)*x + (13*n^2 + 49*n - 157)*x^2 + (2*n^2 + 8*n - 101)*x^3 - 25*x^4 - 2*x^5)*deriv(v[i - 1], x))/(2*n^3 - 26*n - 42 + (4*n^3 + 18*n^2 + 12*n)*x - (18*n + 24)*x^2 - (4*n + 6)*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-dfab35176bd08e72","steps":["x \u21a6 x+1","D+1"],"edges":["e-23e971f1ed1ce0e7","e-27ea45dc0839267a"],"text":"A207606 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A207606 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}