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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":"05b82c720eecf8b2aeeadd3ca3e1d9de32a2c485d87e4d43e04a5a3b006942c4","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-6ffee7a877aa82b8","role":"sequence","name":"basis_map_A007318 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,             816,              324;\n  1:           4913,           20808,            16524,             3456;\n  2:          83521,          471648,           561816,           235008,            32400;\n  3:        1419857,        10022520,         15918120,          9987840,          2754000,           279936;\n  4:       24137569,       204459408,        405912060,        339586560,        140454000,         28553472,          2286144;\n  5:      410338673,      4055111592,       9660707028,      10102700160,       5571342000,       1698931584,        272051136,        17915904;\n  6:     6975757441,     78785025216,     218976025968,     274793444352,     189425628000,      77018231808,      18499477248,      2436562944,       136048896;\n  7:   118587876497,   1506763607256,    4786190281872,    7007232830976,    5796424216800,    2945947366656,     943473339648,    186397065216,     20815481088,     1007769600;\n  8:  2015993900449,  28461090359280,  101706543489780,  170175654466560,  164232019476000,  100162210466304,   40097616935040,  10562500362240,   1769315892480,   171320832000,    7316407296;\n  9: 34271896307633, 532222389718536, 2113235959176540, 3977855923155840, 4387341091716000, 3121722226199808, 1499650873370496, 493796891934720, 110287357297920, 16018497792000, 1368168164352, 52242776064;\n","recurrence_sentence":"The row polynomials satisfy (289+(306*n+510)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (4913+(5202*n+15606)*x+(612*n^2+5508*n+10404)*x^2+(216*n^2+1296*n+1944)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 816*x + 324*x^2.","mathematica":"P[0, x_] := 289 + 816*x + 324*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4913 + (5202*n + 15606)*x + (612*n^2 + 5508*n + 10404)*x^2 + (216*n^2 + 1296*n + 1944)*x^3)*P[n - 1, x])/(289 + (306*n + 510)*x + (36*n^2 + 144*n + 144)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 816*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4913 + (5202*n + 15606)*x + (612*n^2 + 5508*n + 10404)*x^2 + (216*n^2 + 1296*n + 1944)*x^3)*v[i - 1])/(289 + (306*n + 510)*x + (36*n^2 + 144*n + 144)*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-1f2dec04776afb2b","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-653ba39db5056185","e-6aff1710470ffa95"],"text":"A038320 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A038320 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}