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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":"1559f8148798f8e60e516440f0e17fc04304230e0d0296df4fd2951b97108e8f","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-2b477f2aa77e3efc","role":"sequence","name":"Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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:  1;\n  2:  4,    6;\n  3:  9,   46,     45;\n  4: 16,  172,    528,     420;\n  5: 25,  460,   2850,    6756,     4725;\n  6: 36, 1010,  10440,   48556,    97620,     62370;\n  7: 49, 1946,  30135,  229516,   886655,   1585242,    945945;\n  8: 64, 3416,  73920,  832216,  5190080,  17545032,  28687680,  16216200;\n  9: 81, 5592, 161028, 2507736, 22874670, 123422664, 376664148, 573576840, 310134825;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = -(2*n-1+(8*n^2-16*n+6)*x+(8*n^3-28*n^2+30*n-9)*x^2)*P(n-2,x) + (4*n-4+(8*n^2-16*n+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 + (8*n^2 - 16*n + 6)*x + (8*n^3 - 28*n^2 + 30*n - 9)*x^2)*P[n - 2, x] + (4*n - 4 + (8*n^2 - 16*n + 6)*x)*P[n - 1, x])/(2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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