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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":"b2ecc111c2fc48653cd79f32bbad4d16b5787cb0d30bf050c7f7bd893d4e800d","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-b4785bc61afd940f","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see 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:     1;\n  1:     7,      4;\n  2:    26,     34,       8;\n  3:    87,    186,     108,      16;\n  4:   277,    830,     808,     296,      32;\n  5:   854,   3302,    4584,    2824,     752,      64;\n  6:  2573,  12204,   22140,   19536,    8736,    1824,    128;\n  7:  7619,  42848,   96252,  111488,   71440,   24992,   4288,    256;\n  8: 22258, 144842,  388528,  559528,  471280,  236064,  67712,   9856,   512;\n  9: 64323, 475542, 1484184, 2561448, 2686896, 1769184, 725760, 176256, 22272, 1024;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-120*n^2+52*n+34+(64*n^3-192*n^2+32*n+120)*x-(144*n-168)*x^2-(64*n-64)*x^3)*P(n,x) = (24*n^3-168*n^2-170*n+188+(112*n^3-256*n^2-484*n+724)*x+(64*n^3-64*n^2-584*n+848)*x^2-(304*n-400)*x^3-(64*n-64)*x^4)*P(n-1,x) + (20*n^2-80*n-205+(128*n^2-272*n-972)*x+(208*n^2-256*n-1656)*x^2+(64*n^2-64*n-1280)*x^3-464*x^4-64*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((24*n^3 - 168*n^2 - 170*n + 188 + (112*n^3 - 256*n^2 - 484*n + 724)*x + (64*n^3 - 64*n^2 - 584*n + 848)*x^2 - (304*n - 400)*x^3 - (64*n - 64)*x^4)*P[n - 1, x] + (20*n^2 - 80*n - 205 + (128*n^2 - 272*n - 972)*x + (208*n^2 - 256*n - 1656)*x^2 + (64*n^2 - 64*n - 1280)*x^3 - 464*x^4 - 64*x^5)*D[P[n - 1, x], x])/(16*n^3 - 120*n^2 + 52*n + 34 + (64*n^3 - 192*n^2 + 32*n + 120)*x - (144*n - 168)*x^2 - (64*n - 64)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((24*n^3 - 168*n^2 - 170*n + 188 + (112*n^3 - 256*n^2 - 484*n + 724)*x + (64*n^3 - 64*n^2 - 584*n + 848)*x^2 - (304*n - 400)*x^3 - (64*n - 64)*x^4)*v[i - 1] + (20*n^2 - 80*n - 205 + (128*n^2 - 272*n - 972)*x + (208*n^2 - 256*n - 1656)*x^2 + (64*n^2 - 64*n - 1280)*x^3 - 464*x^4 - 64*x^5)*deriv(v[i - 1], x))/(16*n^3 - 120*n^2 + 52*n + 34 + (64*n^3 - 192*n^2 + 32*n + 120)*x - (144*n - 168)*x^2 - (64*n - 64)*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-353fb5a029e4e669","steps":["basis A013609","D+1"],"edges":["e-3748e9c6bfc37784","e-08021cf251922a5e"],"text":"A207607 \u2192 basis A013609 \u2192 D+1"},"headline":"A207607 \u2192 basis A013609 \u2192 D+1","chain":null}