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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":"2bd914279971211e5680ee52bec1a4cba6936d4578e6c82cfc2c07b649dc5a4b","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-649aa8398fb69113","role":"sequence","name":"basis_map_A013609 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,             2128,              1764;\n  1:            6859,            60648,            100548,              43904;\n  2:          130321,          1536416,           3820824,            3336704,             960400;\n  3:         2476099,         36489880,         120992760,          158493440,           91238000,           19361664;\n  4:        47045881,        831969264,        3448293660,         6022750720,         5200566000,         2207229696,          368947264;\n  5:       893871739,      18441985352,       91724611356,       200256461440,       230558426000,       146780774784,        49069986112,         6746464256;\n  6:     16983563041,     400454539072,     2323690154352,      6087796427776,      8761220188000,      7436892589056,      3729318944512,      1025462566912,       119538913536;\n  7:    322687697779,    8559715772664,    56764430913456,    173502198191616,    299633730429600,    317927158182144,    212571179837184,     87677049470976,     20441154214656,     2066104678400;\n  8:   6131066257801,  180705110756240,  1348155234194580,   4709345379486720,   9488401463604000,  12081232010921472,  10097131042266240,   5552879799828480,   1941909650392320,   392559888896000,   34999813252096;\n  9: 116490258898219, 3776736814805416, 31307160438518580, 123031648039090560, 283296557984748000, 420829581713764608, 422060077566728832, 290137969541038080, 135286372310664960, 41022508389632000, 7314960969688064, 583137384431616;\n","recurrence_sentence":"The row polynomials satisfy (361+(798*n+1330)*x+(196*n^2+784*n+784)*x^2)*P(n,x) = (6859+(15162*n+45486)*x+(3724*n^2+33516*n+63308)*x^2+(2744*n^2+16464*n+24696)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 2128*x + 1764*x^2.","mathematica":"P[0, x_] := 361 + 2128*x + 1764*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((6859 + (15162*n + 45486)*x + (3724*n^2 + 33516*n + 63308)*x^2 + (2744*n^2 + 16464*n + 24696)*x^3)*P[n - 1, x])/(361 + (798*n + 1330)*x + (196*n^2 + 784*n + 784)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 361 + 2128*x + 1764*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6859 + (15162*n + 45486)*x + (3724*n^2 + 33516*n + 63308)*x^2 + (2744*n^2 + 16464*n + 24696)*x^3)*v[i - 1])/(361 + (798*n + 1330)*x + (196*n^2 + 784*n + 784)*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":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-20f4338b8f3f0a8d","e-2c33a2edb4e05388"],"text":"A038333 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038333 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}