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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":"ac2d9c903061ed907046c75b4892a7084c638110f9061678df6e62f99fef5a45","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-54fca3fdfcf03464","role":"sequence","name":"basis_map_A007318 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,             1064,              441;\n  1:            6859,            30324,            25137,              5488;\n  2:          130321,           768208,           955206,            417088,             60025;\n  3:         2476099,         18244940,         30248190,          19811680,           5702375,            605052;\n  4:        47045881,        415984632,        862073415,         752843840,         325035375,          68975928,          5764801;\n  5:       893871739,       9220992676,      22931152839,       25032057680,       14409901625,        4586899212,        766718533,         52706752;\n  6:     16983563041,     200227269536,     580922538588,      760974553472,      547576261750,      232402893408,      58270608508,       8011426304,       466948881;\n  7:    322687697779,    4279857886332,   14191107728364,    21687774773952,    18727108151850,     9935223693192,    3321424684956,     684976948992,     79848258651,     4035360700;\n  8:   6131066257801,   90352555378120,  337038808548645,   588668172435840,   593025091475250,   377538500341296,  157767672535410,   43381873436160,   7585584571845,   766718533000,   34179505129;\n  9: 116490258898219, 1888368407402708, 7826790109629645, 15378956004886320, 17706034874046750, 13150924428555144, 6594688711980138, 2266702887039360, 528462391838535, 80122086698500, 7143516571961, 284735050992;\n","recurrence_sentence":"The row polynomials satisfy (361+(399*n+665)*x+(49*n^2+196*n+196)*x^2)*P(n,x) = (6859+(7581*n+22743)*x+(931*n^2+8379*n+15827)*x^2+(343*n^2+2058*n+3087)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 1064*x + 441*x^2.","mathematica":"P[0, x_] := 361 + 1064*x + 441*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((6859 + (7581*n + 22743)*x + (931*n^2 + 8379*n + 15827)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*P[n - 1, x])/(361 + (399*n + 665)*x + (49*n^2 + 196*n + 196)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 361 + 1064*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6859 + (7581*n + 22743)*x + (931*n^2 + 8379*n + 15827)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*v[i - 1])/(361 + (399*n + 665)*x + (49*n^2 + 196*n + 196)*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-5cf6acfdac607d6d","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-737551651b8dc504","e-39f68850017665be"],"text":"A038333 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A038333 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}