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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":"66affc823eed3d1366723f82d3d97c687c83c054c251572acaa4afb43970dc19","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-c21cdce554fb6206","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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,            1224,              972;\n  1:           4913,           31212,            49572,             23328;\n  2:          83521,          707472,          1685448,           1586304,            524880;\n  3:        1419857,        15033780,         47754360,          67417920,          44614800,           11337408;\n  4:       24137569,       306689112,       1217736180,        2292209280,        2275354800,         1156415616,          238085568;\n  5:      410338673,      6082667388,      28982121084,       68193226080,       90255740400,        68806729152,        28332182592,         4897760256;\n  6:     6975757441,    118177537824,     656928077904,     1854855749376,     3068695173600,      3119238388224,      1926588416256,       666095394816,       99179645184;\n  7:   118587876497,   2260145410884,   14358570845616,    47298821609088,    93902072312160,    119310868349568,     98256009229056,     50956297703424,    15174485713152,     1983592903680;\n  8:  2015993900449,  42691635538920,  305119630469340,  1148685667649280,  2660558715511200,   4056569523885312,   4175880392234880,   2887523536527360,  1289831285617920,   337210793625600,   39275139492864;\n  9: 34271896307633, 798333584577804, 6339707877529620, 26850527481301920, 71074925685799200, 126429750161092224, 156177926669584512, 134991725332654080, 80399483470183680, 31529209203993600, 7344451085165568, 771220920950784;\n","recurrence_sentence":"The row polynomials satisfy (17+(18*n+36)*x)*P(n,x) = (289+(306*n+1224)*x+(324*n+972)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1224*x + 972*x^2.","mathematica":"P[0, x_] := 289 + 1224*x + 972*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((289 + (306*n + 1224)*x + (324*n + 972)*x^2)*P[n - 1, x])/(17 + (18*n + 36)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 1224*x + 972*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((289 + (306*n + 1224)*x + (324*n + 972)*x^2)*v[i - 1])/(17 + (18*n + 36)*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-1f2dec04776afb2b","steps":["basis A013610","\u03b8+1"],"edges":["e-572c9115d911d6cf","e-732ee758aabf28ea"],"text":"A038320 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038320 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}