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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":"a7b78b05b2437ffff8168f3842f7210dea4bdbe4d0c1d5cd16c216a699a6bf96","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-43a1e62341e55229","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+11x)^n.","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:          144,           2112,             4356;\n  1:         1728,          38016,           156816,            170368;\n  2:        20736,         608256,          3763584,           8177664,           5856400;\n  3:       248832,        9123840,         75271680,         245329920,         351384000,          185530752;\n  4:      2985984,      131383296,       1354890240,        5887918080,       12649824000,        13358214144,         5555615296;\n  5:     35831808,     1839366144,      22762156032,      123646279680,      354195072000,       561044994048,       466671684864,        159638904832;\n  6:    429981696,    25225592832,     364194496512,     2374008569856,     8500681728000,     17953439809536,     22400240873472,      15325334863872,       4444945756416;\n  7:   5159780352,   340545503232,    5619000803328,    42732154257408,   183614725324800,    484742874857472,    806408671444992,     827568082649088,     480054141692928,    120726921779200;\n  8:  61917364224,  4540606709760,   84285012049920,   732551215841280,  3672294506496000,  11633828996579328,  24192260143349760,   33102723305963520,   28803248501575680,  14487230613504000,   3213750657762304;\n  9: 743008370688, 59936008568832, 1236180176732160, 12087095061381120, 69248982122496000, 255944237924745216, 638675667784433664, 1092389869096796160, 1267342934069329920, 956157220491264000, 424215086824624128, 84141835403231232;\n","recurrence_sentence":"The row polynomials satisfy (36+(198*n+330)*x+(121*n^2+484*n+484)*x^2)*P(n,x) = (432+(2376*n+7128)*x+(1452*n^2+13068*n+24684)*x^2+(2662*n^2+15972*n+23958)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 2112*x + 4356*x^2.","mathematica":"P[0, x_] := 144 + 2112*x + 4356*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((432 + (2376*n + 7128)*x + (1452*n^2 + 13068*n + 24684)*x^2 + (2662*n^2 + 15972*n + 23958)*x^3)*P[n - 1, x])/(36 + (198*n + 330)*x + (121*n^2 + 484*n + 484)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 144 + 2112*x + 4356*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((432 + (2376*n + 7128)*x + (1452*n^2 + 13068*n + 24684)*x^2 + (2662*n^2 + 15972*n + 23958)*x^3)*v[i - 1])/(36 + (198*n + 330)*x + (121*n^2 + 484*n + 484)*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-ec89fb3fc20684fc","steps":["x \u21a6 x/(x+1)","(\u03b8+1)\u00b2"],"edges":["e-492aa71fabef2dc6","e-4a7b02913011f637"],"text":"A038336 \u2192 x \u21a6 x/(x+1) \u2192 (\u03b8+1)\u00b2"},"headline":"A038336 \u2192 x \u21a6 x/(x+1) \u2192 (\u03b8+1)\u00b2","chain":null}