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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":"f16e20163bfd84085a6c3e2705015e8dbfe54756f9e300c00631ec9e801fe9bb","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-735e311cbe9a718e","role":"sequence","name":"Triangle T(n, k) read by rows: T(n, k) = (Sum_{i=0..k} (-1)^i * binomial(k, i) * (n+i)^n) * (-1)^k / k! for 0 <= k <= n.","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:          9,          7,         1;\n  3:         64,         61,        15,         1;\n  4:        625,        671,       217,        26,        1;\n  5:       7776,       9031,      3465,       565,       40,       1;\n  6:     117649,     144495,     62401,     12410,     1220,      57,      1;\n  7:    2097152,    2685817,   1265607,    289821,    35350,    2324,     77,    1;\n  8:   43046721,   56953279,  28702801,   7309722,  1041831,   85974,   4046,  100,   1;\n  9: 1000000000, 1357947691, 721942485, 199833565, 32080860, 3125451, 186186, 6582, 126, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+1+x)*P(n-1,x) + (n+1+2*x)*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + 1 + x)*P[n - 1, x] + (n + 1 + 2*x)*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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