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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":"7896a3fe5c69c6d631ff37dbe4188aaee671f89535e50b39cd7a6370774cc796","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-74f9a0f107d622a7","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.","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:   8,     9,       1;\n  3:  20,    57,      22,        1;\n  4:  40,   381,     288,       43,        1;\n  5:  70,  1591,    3044,     1037,       74,        1;\n  6: 112,  4969,   28028,    18249,     2984,      117,       1;\n  7: 168, 12837,  164786,   264085,    84492,     7337,     174,     1;\n  8: 240, 29005,  709216,  3077519,  1869532,   317909,   16056,   247,   1;\n  9: 330, 59299, 2458912, 23385013, 33862806, 10502655, 1016204, 32117, 338, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^5-3*n^4-n^3+5*n^2+n-1)*x^2*P(n-4,x) - ((2*n^3-4*n-2)*x+(2*n^3-2*n^2-2*n)*x^2)*P(n-3,x) - (n+1-(2*n^3-6*n)*x+(n-1)*x^2)*P(n-2,x) + (2*n+(2*n-2)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 8 + 9*x + x^2, P(3,x) = 20 + 57*x + 22*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 8 + 9*x + x^2;\nP[3, x_] := 20 + 57*x + 22*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 3*n^4 - n^3 + 5*n^2 + n - 1)*x^2*P[n - 4, x] - ((2*n^3 - 4*n - 2)*x + (2*n^3 - 2*n^2 - 2*n)*x^2)*P[n - 3, x] - (n + 1 - (2*n^3 - 6*n)*x + (n - 1)*x^2)*P[n - 2, x] + (2*n + (2*n - 2)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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