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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":"cf58e01aca1d2227e2ec0ea9f8b13e478614576fd3dc58e3ae651dd6559f9868","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-1397bdceeefc7b3d","role":"sequence","name":"Triangle read by rows T(n, k) = binomial(2*n, k) * binomial(3*n - k, 2*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:        0;\n  1:        2;\n  2:       20,        12;\n  3:      168,       210,        60;\n  4:     1320,      2520,      1512,       280;\n  5:    10010,     25740,     23760,      9240,      1260;\n  6:    74256,    240240,    300300,    180180,     51480,      5544;\n  7:   542640,   2116296,   3341520,   2722720,   1201200,    270270,    24024;\n  8:  3922512,  17907120,  34186320,  35271600,  21162960,   7351344,  1361360,  102960;\n  9: 28120950, 147094200, 329491008, 411863760, 313374600, 148140720, 42325920, 6651216, 437580;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-10*n^2+8*n-2)*P(n,x) = (27*n^3-54*n^2+33*n-6+(24*n^3-55*n^2+35*n-6)*x+(n^3-3*n^2+2*n)*x^2)*P(n-1,x) - ((9*n^2-9*n+2)*x+(10*n^2-10*n+2)*x^2+(n^2-n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((27*n^3 - 54*n^2 + 33*n - 6 + (24*n^3 - 55*n^2 + 35*n - 6)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 1, x] - ((9*n^2 - 9*n + 2)*x + (10*n^2 - 10*n + 2)*x^2 + (n^2 - n)*x^3)*D[P[n - 1, x], x])/(4*n^3 - 10*n^2 + 8*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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