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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":"f6b354b9132f910d89fde99980ab64c7c756e6b0a700c1e8c9f57d7a591b8022","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-eed4f744ce063213","role":"sequence","name":"Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^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:  3,   4;\n  2:  6,  26,   36,    16;\n  3:  9,  66,  189,   264,    180,      48;\n  4: 12, 124,  540,  1284,   1800,    1488,     672,     128;\n  5: 15, 200, 1170,  3940,   8415,   11820,   10920,    6400,     2160,      320;\n  6: 18, 294, 2160,  9420,  27090,   53934,   75852,   75360,    51840,    23520,     6336,      768;\n  7: 21, 406, 3591, 19236,  69615,  179718,  340473,  479248,   501228,   384720,   210672,    77952,    17472,    1792;\n  8: 24, 536, 5544, 35224, 153720,  488040, 1164408, 2125832,  2994192,  3253600,  2705472,  1690752,   768768,  240128,   46080,   4096;\n  9: 27, 684, 8100, 59544, 304290, 1147104, 3302964, 7419528, 13163067, 18548820, 20761488, 18353664, 12658464, 6668928, 2592000, 700416, 117504, 9216;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n+3*n*x+2*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 3*n*x + 2*n*x^2)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + 3*n*x + 2*n*x^2)*v[i - 1])/(n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-eed4f744ce063213","steps":["D"],"edges":["e-f63690695a9b626c"],"text":"A200536 \u2192 D"},"headline":"A200536 \u2192 D","chain":null}