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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":"24c13c56886349b793bf0dd9d97521e872c25732ef988807db8f1b67c26b6289","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-806f02861480e04e","role":"sequence","name":"Exponential Riordan array [exp(sinh(x)*exp(x)), sinh(x)*exp(x)].","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       2,       1;\n   3:       7,       6,        1;\n   4:      30,      37,       12,       1;\n   5:     153,     244,      118,      20,       1;\n   6:     898,    1741,     1120,     290,      30,      1;\n   7:    5919,   13442,    10771,    3780,     605,     42,     1;\n   8:   43102,  111833,   106988,   47551,   10430,   1127,    56,    1;\n   9:  342673,  997416,  1106540,  595896,  167286,  24920,  1932,   72,  1;\n  10: 2946946, 9489145, 11953248, 7560820, 2598036, 498414, 53424, 3108, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*x+x^2)*P(n-2,x) + (2+4*x+2*x^2)*d/dx P(n-2,x) + 2*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*x + x^2)*P[n - 2, x] + (2 + 4*x + 2*x^2)*D[P[n - 2, x], x] + 2*P[n - 1, x] + (2 + 2*x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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