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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":"904da4a3909a2c1d958ca2b80762671c77a1f2c308f33c53611895fbbf1984c9","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-97b51734d39ede6c","role":"sequence","name":"Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph 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:  1;\n  2:  2,   1;\n  3:  4,   3;\n  4:  7,  12,    3;\n  5: 11,  40,   15;\n  6: 16, 105,   90,   15;\n  7: 22, 231,  420,  105;\n  8: 29, 448, 1470,  840, 105;\n  9: 37, 792, 4158, 5040, 945;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+4+(4*n-10)*x+4*x^2)*P(n,x) = (n^2-n+2+(n^3-4*n^2+9*n-12)*x+(4*n^2-10*n+10)*x^2+(4*n-4)*x^3)*P(n-1,x) - ((2*n^2-2*n+4)*x^2+(8*n-4)*x^3+8*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 - n + 2 + (n^3 - 4*n^2 + 9*n - 12)*x + (4*n^2 - 10*n + 10)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] - ((2*n^2 - 2*n + 4)*x^2 + (8*n - 4)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(n^2 - 3*n + 4 + (4*n - 10)*x + 4*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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