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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":"add2c4f292ebf44f22e3de2e014344cc4b688b88227f32ab66c66211b483c6d7","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-8eb9bb3a09f0b7fd","role":"sequence","name":"Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (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:   6,    8;\n  1:  12,   40,    24;\n  2:  20,  120,   168,     64;\n  3:  30,  280,   672,    576,     160;\n  4:  42,  560,  2016,   2880,    1760,     384;\n  5:  56, 1008,  5040,  10560,   10560,    4992,     896;\n  6:  72, 1680, 11088,  31680,   45760,   34944,   13440,    2048;\n  7:  90, 2640, 22176,  82368,  160160,  174720,  107520,   34816,   4608;\n  8: 110, 3960, 41184, 192192,  480480,  698880,  609280,  313344,  87552,  10240;\n  9: 132, 5720, 72072, 411840, 1281280, 2376192, 2741760, 1984512, 875520, 215040, 22528;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+5*n+3)*P(n,x) = (2*n^2+9*n+9+(2*n^2+10*n+12)*x)*P(n-1,x) + ((4*n+8)*x+(2*n+4)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + 8*x.","mathematica":"P[0, x_] := 6 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 9*n + 9 + (2*n^2 + 10*n + 12)*x)*P[n - 1, x] + ((4*n + 8)*x + (2*n + 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 + 5*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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