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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":"d6a482993a908fdbce58a0f71241e46e92dff430d2b00b11ff4da0293fdcd7ff","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-19327cf22807de6e","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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:             529,              2024,               1089;\n  1:           12167,             69828,              75141,              21296;\n  2:          279841,           2141392,            3456486,            1959232,             366025;\n  3:         6436343,          61565020,          132498630,          112655840,           42092875,            5797836;\n  4:       148035889,        1699194552,         4571202735,         5182168640,         2904408375,          800101368,           86806489;\n  5:      3404825447,       45595053812,       147192728067,       208582287760,       155869916125,        64408160124,        13975844729,         1247178944;\n  6:     78310985281,     1198498557344,      4513910327388,      7675828189568,      7170016141750,      3950367154272,      1285777715068,       229480925696,       17363069361;\n  7:   1801152661463,    31011150171276,    133482776824188,    264816072540096,    296838668268450,    204431500233576,     88718662339692,     23751275809536,     3594155357727,     235794769100;\n  8:  41426511213649,   792507171043720,   3837629833695405,   8701099526317440,  11378815616957250,   9403849010744496,   5101323084532290,   1820931145397760,   413327866138605,   54232796893000,   3138428376721;\n  9: 952809757913927, 20050431427406116, 107880038658326385, 275172272519789040, 411262907298597750, 396528966619726248, 258126948077333874, 115173894946408320, 34857316711022355, 6860448806964500, 794022379310413, 41084880567984;\n","recurrence_sentence":"The row polynomials satisfy (529+(759*n+1265)*x+(121*n^2+484*n+484)*x^2)*P(n,x) = (12167+(17457*n+52371)*x+(2783*n^2+25047*n+47311)*x^2+(1331*n^2+7986*n+11979)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 2024*x + 1089*x^2.","mathematica":"P[0, x_] := 529 + 2024*x + 1089*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((12167 + (17457*n + 52371)*x + (2783*n^2 + 25047*n + 47311)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*P[n - 1, x])/(529 + (759*n + 1265)*x + (121*n^2 + 484*n + 484)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 529 + 2024*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12167 + (17457*n + 52371)*x + (2783*n^2 + 25047*n + 47311)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*v[i - 1])/(529 + (759*n + 1265)*x + (121*n^2 + 484*n + 484)*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-344e4bfa9933088a","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-347b6efdbfa5233f","e-06cb9a8e802e0462"],"text":"A038337 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A038337 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}