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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":"56ac11f499810e3a81479bca18d96130d9667c0a947659994bd4241d04d3205d","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-4598f43d5daedb7a","role":"sequence","name":"reversal of Riordan array (1, 3+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:     3;\n   3:    10,     1;\n   4:    33,     6;\n   5:   108,    29,    1;\n   6:   351,   126,    9;\n   7:  1134,   513,   57,   1;\n   8:  3645,  1998,  306,  12;\n   9: 11664,  7533, 1485,  94,  1;\n  10: 37179, 27702, 6723, 600, 15;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3+9*n^2-48*n+36+(12*n^2-27*n+15)*x+(12*n-12)*x^2)*P(n,x) = (9*n^3+36*n^2-171*n+126+(2*n^3+40*n^2-107*n+65)*x+(8*n^2+18*n-32)*x^2+(8*n-16)*x^3)*P(n-1,x) - ((9*n^2+54*n-63)*x+(4*n^2+60*n-1)*x^2+(16*n+48)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^3 + 36*n^2 - 171*n + 126 + (2*n^3 + 40*n^2 - 107*n + 65)*x + (8*n^2 + 18*n - 32)*x^2 + (8*n - 16)*x^3)*P[n - 1, x] - ((9*n^2 + 54*n - 63)*x + (4*n^2 + 60*n - 1)*x^2 + (16*n + 48)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(3*n^3 + 9*n^2 - 48*n + 36 + (12*n^2 - 27*n + 15)*x + (12*n - 12)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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