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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":"b2a9b43b101d856d04bd9ab8e150eacd71c92d8f1111a3c9bc5cf0e30c062917","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-07e5356ad37f61ea","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'.","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:  4,    5,     1;\n  3:  7,   18,     9,     1;\n  4: 11,   52,    51,    14,     1;\n  5: 16,  125,   220,   115,    20,     1;\n  6: 22,  261,   765,   695,   225,    27,    1;\n  7: 29,  490,  2226,  3325,  1820,   399,   35,    1;\n  8: 37,  848,  5614, 13076, 11515,  4172,  658,   44,  1;\n  9: 46, 1377, 12636, 43554, 59346, 33831, 8652, 1026, 54, 1;\n","recurrence_sentence":"The row polynomials satisfy -(n^2-n+2-(n^2-n+8)*x-(2*n-6)*x^2-2*x^3)*P(n,x) = -(n^2+n+2+(n^3-2*n^2-n-12)*x+(n^2-7*n+6)*x^2-2*x^4)*P(n-1,x) + ((2*n^2+2*n+4)*x^2+(4*n-4)*x^3+4*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 - 2*n^2 - n - 12)*x + (n^2 - 7*n + 6)*x^2 - 2*x^4)*P[n - 1, x] + ((2*n^2 + 2*n + 4)*x^2 + (4*n - 4)*x^3 + 4*x^4)*D[P[n - 1, x], x])/(-n^2 + n - 2 + (n^2 - n + 8)*x + (2*n - 6)*x^2 + 2*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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