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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":"c94520740ef1bd0f89d1623ec459d32d07eae1c0764b83f18708830f0cff2802","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-522def1e8ebf49cd","role":"sequence","name":"Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; 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:  1;\n  1:  2,   1;\n  2:  5,   6,    1;\n  3:  8,  25,   12,     1;\n  4: 11,  62,   74,    20,     1;\n  5: 14, 117,  262,   170,    30,     1;\n  6: 17, 190,  657,   810,   335,    42,    1;\n  7: 20, 281, 1340,  2615,  2058,   595,   56,    1;\n  8: 23, 390, 2392,  6584,  8322,  4564,  980,   72,  1;\n  9: 26, 517, 3894, 14040, 25436, 22590, 9156, 1524, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-7*n^2+4*n-(20*n^3-51*n^2+42*n)*x+(32*n^3-144*n^2+205*n-63)*x^2+(64*n^2-196*n+105)*x^3+(32*n-24)*x^4)*P(n,x) = (3*n^3-4*n^2+n-(2*n^3+16*n^2+4*n)*x-(52*n^3-133*n^2+59*n-27)*x^2+(96*n^3-368*n^2+356*n-108)*x^3+(192*n^2-516*n+313)*x^4+(96*n-88)*x^5)*P(n-1,x) - ((3*n^2-n)*x+(13*n^2+7*n-3)*x^2-(80*n^2-90*n+2)*x^3+(64*n^2-224*n+149)*x^4+(128*n-208)*x^5+64*x^6)*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[((3*n^3 - 4*n^2 + n - (2*n^3 + 16*n^2 + 4*n)*x - (52*n^3 - 133*n^2 + 59*n - 27)*x^2 + (96*n^3 - 368*n^2 + 356*n - 108)*x^3 + (192*n^2 - 516*n + 313)*x^4 + (96*n - 88)*x^5)*P[n - 1, x] - ((3*n^2 - n)*x + (13*n^2 + 7*n - 3)*x^2 - (80*n^2 - 90*n + 2)*x^3 + (64*n^2 - 224*n + 149)*x^4 + (128*n - 208)*x^5 + 64*x^6)*D[P[n - 1, x], x])/(3*n^3 - 7*n^2 + 4*n - (20*n^3 - 51*n^2 + 42*n)*x + (32*n^3 - 144*n^2 + 205*n - 63)*x^2 + (64*n^2 - 196*n + 105)*x^3 + (32*n - 24)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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