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(f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"fffcc542ceefd5f8eded93d2059b17342e02df2722beeb5fab549d05b7da75e1","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-4af84b2e19878bab","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.","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:            289,             680,              225;\n  1:           4913,           17340,            11475,             2000;\n  2:          83521,          393040,           390150,           136000,            15625;\n  3:        1419857,         8352100,         11054250,          5780000,          1328125,           112500;\n  4:       24137569,       170382840,        281883375,        196520000,         67734375,         11475000,          765625;\n  5:      410338673,      3379259660,       6708824325,       5846470000,       2686796875,        682762500,        91109375,         5000000;\n  6:     6975757441,     65654187680,     152066684700,     159023984000,      91351093750,      30951900000,      6195437500,       680000000,       31640625;\n  7:   118587876497,   1255636339380,    3323743251300,    4055111592000,    2795343468750,    1183910175000,    315967312500,     52020000000,     4841015625,     195312500;\n  8:  2015993900449,  23717575299400,   70629544090125,   98481281520000,   79201398281250,   40252945950000,  13428610781250,   2947800000000,   411486328125,   33203125000,   1181640625;\n  9: 34271896307633, 443518658098780, 1467524971650375, 2301999955530000, 2115808782656250, 1254550148775000, 502230043218750, 137809650000000, 25649314453125, 3104492187500, 220966796875, 7031250000;\n","recurrence_sentence":"The row polynomials satisfy (289+(255*n+425)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (4913+(4335*n+13005)*x+(425*n^2+3825*n+7225)*x^2+(125*n^2+750*n+1125)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 680*x + 225*x^2.","mathematica":"P[0, x_] := 289 + 680*x + 225*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4913 + (4335*n + 13005)*x + (425*n^2 + 3825*n + 7225)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*P[n - 1, x])/(289 + (255*n + 425)*x + (25*n^2 + 100*n + 100)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 680*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4913 + (4335*n + 13005)*x + (425*n^2 + 3825*n + 7225)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*v[i - 1])/(289 + (255*n + 425)*x + (25*n^2 + 100*n + 100)*x^2)); 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-a5302200d285579d","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-bdf366c5a3f3b64d","e-71faf430ae512263"],"text":"A038331 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A038331 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}