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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":"5f941fdca4e61a885cf5f0e66c6bba190c09eef29b0d7dc0104e8bca37740a9f","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-0990b5be9f344750","role":"sequence","name":"Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","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:  2;\n  2:  5,    8;\n  3:  9,   38,    24;\n  4: 14,  110,   195,     64;\n  5: 20,  250,   855,    844,    160;\n  6: 27,  490,  2730,   5404,   3325,     384;\n  7: 35,  868,  7140,  23604,  30345,   12354,    896;\n  8: 44, 1428, 16254,  80724, 177870,  158010,  44135,   2048;\n  9: 54, 2220, 33390, 232428, 783300, 1225230, 781515, 153368, 4608;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n+1+(2*n^2-2)*x+(n^3-n)*x^2)*P(n-2,x) + ((n+1)*x^2+(n^2+n)*x^3)*d/dx P(n-2,x) + (2*n+1+2*n^2*x)*P(n-1,x) - n*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (2*n^2 - 2)*x + (n^3 - n)*x^2)*P[n - 2, x] + ((n + 1)*x^2 + (n^2 + n)*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 2*n^2*x)*P[n - 1, x] - n*x^2*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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