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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":"8244e6e7ed0d38974655c6c089f0dc2590310aaca2f8d7a8cdd0ed86d624321d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9d2e609ab19d361f","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (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:  5,   6,    1;\n  3:  7,  18,    9,    1;\n  4:  9,  40,   40,   12,    1;\n  5: 11,  70,  120,   70,   15,    1;\n  6: 13, 108,  276,  268,  108,   18,    1;\n  7: 15, 154,  532,  770,  504,  154,   21,   1;\n  8: 17, 208,  912, 1808, 1744,  848,  208,  24,  1;\n  9: 19, 270, 1440, 3678, 4860, 3438, 1320, 270, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^3-7*n^2+7*n-2-(2*n^3-9*n^2+20*n-13)*x-(5*n^2-16*n+11)*x^2-(n^2+2*n-3)*x^3-(n-1)*x^4)*P(n,x) = -(2*n^3-5*n^2+n+2+(6*n^3-17*n^2-6*n+17)*x+(15*n^2-36*n+21)*x^2+(3*n^2+10*n-13)*x^3+(3*n-3)*x^4)*P(n-1,x) + ((2*n^2-n-1)*x+(12*n^2-8)*x^2+(2*n^2+36*n-7)*x^3+(12*n+35)*x^4+(n+12)*x^5+x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 5*n^2 + n + 2 + (6*n^3 - 17*n^2 - 6*n + 17)*x + (15*n^2 - 36*n + 21)*x^2 + (3*n^2 + 10*n - 13)*x^3 + (3*n - 3)*x^4)*P[n - 1, x] + ((2*n^2 - n - 1)*x + (12*n^2 - 8)*x^2 + (2*n^2 + 36*n - 7)*x^3 + (12*n + 35)*x^4 + (n + 12)*x^5 + x^6)*D[P[n - 1, x], x])/(-2*n^3 + 7*n^2 - 7*n + 2 + (2*n^3 - 9*n^2 + 20*n - 13)*x + (5*n^2 - 16*n + 11)*x^2 + (n^2 + 2*n - 3)*x^3 + (n - 1)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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