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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":"5fd9ab1c04b21c8f8c4dc8740e5cacca895f27dfdf403b6c516070b318efde25","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-6aadffa116a44288","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","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:       0;\n  1:       6;\n  2:      42,       54;\n  3:     240,      666,       324;\n  4:    1248,     5328,      6156,      1620;\n  5:    6144,    35424,     66096,     43740,      7290;\n  6:   29184,   211968,    549504,    607824,    264870,     30618;\n  7:  135168,  1184256,   3939840,   6189696,   4626720,   1439046,    122472;\n  8:  614400,  6303744,  25629696,  52462080,  56687040,  30862944,   7225848,   472392;\n  9: 2752512, 32366592, 155602944, 393569280, 561116160, 448037568, 186647328, 34169688, 1771470;\n","recurrence_sentence":"The row polynomials satisfy -(6*n^3-19*n^2+19*n-6-(6*n^3-9*n^2+3*n)*x)*P(n,x) = -(24*n^3-52*n^2+16*n+12+(30*n^3-117*n^2+51*n+36)*x-(54*n^3-63*n^2-9*n)*x^2)*P(n-1,x) + ((12*n^2-8*n-4)*x+(24*n^2-24*n-12)*x^2-36*n^2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[(-(24*n^3 - 52*n^2 + 16*n + 12 + (30*n^3 - 117*n^2 + 51*n + 36)*x - (54*n^3 - 63*n^2 - 9*n)*x^2)*P[n - 1, x] + ((12*n^2 - 8*n - 4)*x + (24*n^2 - 24*n - 12)*x^2 - 36*n^2*x^3)*D[P[n - 1, x], x])/(-6*n^3 + 19*n^2 - 19*n + 6 + (6*n^3 - 9*n^2 + 3*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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