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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":"352a8a6ad303ad69419d9a08dea168a4d1a5009741e7b1f5158320a0754b98de","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5576233c61c40153","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of descents.","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,  2;\n  2: 1,  9,  4;\n  3: 0, 12, 30,   8;\n  4: 0,  2, 73,  88,   16;\n  5: 0,  0, 30, 330,  240,   32;\n  6: 0,  0,  3, 241, 1260,  624,    64;\n  7: 0,  0,  0,  56, 1414, 4312,  1568,   128;\n  8: 0,  0,  0,   4,  561, 6832, 13664,  3840,  256;\n  9: 0,  0,  0,   0,   90, 4050, 28896, 40896, 9216, 512;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^3-4*n^2+2*n-(32*n^3-120*n^2+148*n-60)*x-(64*n^2-184*n+120)*x^2-(32*n-32)*x^3)*P(n,x) = (n^3-4*n^2+3*n-(28*n^3-84*n^2+62*n-6)*x+(192*n^3-624*n^2+516*n-84)*x^2+(384*n^2-928*n+544)*x^3+(192*n-192)*x^4)*P(n-1,x) - ((2*n^2-2*n)*x-(40*n^2-48*n+4)*x^2+(128*n^2-224*n+72)*x^3+(256*n-256)*x^4+128*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - 4*n^2 + 3*n - (28*n^3 - 84*n^2 + 62*n - 6)*x + (192*n^3 - 624*n^2 + 516*n - 84)*x^2 + (384*n^2 - 928*n + 544)*x^3 + (192*n - 192)*x^4)*P[n - 1, x] - ((2*n^2 - 2*n)*x - (40*n^2 - 48*n + 4)*x^2 + (128*n^2 - 224*n + 72)*x^3 + (256*n - 256)*x^4 + 128*x^5)*D[P[n - 1, x], x])/(-2*n^3 + 4*n^2 - 2*n + (32*n^3 - 120*n^2 + 148*n - 60)*x + (64*n^2 - 184*n + 120)*x^2 + (32*n - 32)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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