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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":"f86f93b627478531729c9f8951ccc6fbf971efa20ff632660f94caf391c98fa7","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-f96c16e0c5fd8334","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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:             361,             1596,              1323;\n  1:            6859,            45486,             75411,              37044;\n  2:          130321,          1152312,           2865618,            2815344,             972405;\n  3:         2476099,         27367410,          90744570,          133728840,           92378475,           24504606;\n  4:        47045881,        623976948,        2586220245,         5081695920,         5265573075,         2793525084,          600362847;\n  5:       893871739,      13831489014,       68793458517,       168966389340,       233440406325,       185769418086,        79848258651,        14408708328;\n  6:     16983563041,     300340904304,     1742767615764,      5136578235936,      8870735440350,      9412317183024,      6068467657476,      2190123665856,       340405734249;\n  7:    322687697779,    6419786829498,    42573323185092,    146392479724176,    303379152059970,    402376559574276,    345902656476132,    187255573430688,     58209380556579,      7942800465810;\n  8:   6131066257801,  135528833067180,  1011116425645935,   3973510163941920,   9607006481899050,  15290309263822488,  16430376182616270,  11859519650610240,   5529891152875005,   1509132088503900,   183478690760211;\n  9: 116490258898219, 2832552611104062, 23480370328888935, 103807953032982660, 286837764959557350, 532612439356483332, 686789724433360086, 619659901744385040, 385249083650292015, 157704303248657550, 38347046368884099, 4203330006506652;\n","recurrence_sentence":"The row polynomials satisfy (19+(21*n+42)*x)*P(n,x) = (361+(399*n+1596)*x+(441*n+1323)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 1596*x + 1323*x^2.","mathematica":"P[0, x_] := 361 + 1596*x + 1323*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((361 + (399*n + 1596)*x + (441*n + 1323)*x^2)*P[n - 1, x])/(19 + (21*n + 42)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 361 + 1596*x + 1323*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((361 + (399*n + 1596)*x + (441*n + 1323)*x^2)*v[i - 1])/(19 + (21*n + 42)*x)); 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-5cf6acfdac607d6d","steps":["basis A013610","\u03b8+1"],"edges":["e-065a527a7a0e90c1","e-def4eb5c13015429"],"text":"A038333 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038333 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}