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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":"5102e1a674bce10251561faa27ed8eb362999a04296bbde88ef98fd121a3e844","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-32cf9641286d8e9f","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(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:            324,             1008,              441;\n  1:           5832,            27216,            23814,             5488;\n  2:         104976,           653184,           857304,           395136,             60025;\n  3:        1889568,         14696640,         25719120,         17781120,           5402250,           605052;\n  4:       34012224,        317447424,        694416240,        640120320,         291721500,         65345616,          5764801;\n  5:      612220032,       6666395904,      17499289248,      20163790080,       12252303000,       4116773808,        726364926,         52706752;\n  6:    11019960576,     137137287168,     419982941952,     580717154304,      441082908000,     197605142784,      52298274672,       7589772288,       466948881;\n  7:   198359290368,    2777030065152,    9719605228032,   15679363166208,    14291086219200,    8003008282752,    2824106832288,     614771555328,     75645718722,     4035360700;\n  8:  3570467226624,   55540601303040,  218691117630720,  403183624273920,   428732586576000,  288108298179072,  127084807452960,   36886293319680,   6808114684980,   726364926000,   34179505129;\n  9: 64268410079232, 1099703905800192, 4811204587875840, 9978794700779520, 12127007448864000, 9507573839909376, 5032558375137216, 1825871519324160, 449335569208680, 71910127674000, 6767542015542, 284735050992;\n","recurrence_sentence":"The row polynomials satisfy (324+(378*n+630)*x+(49*n^2+196*n+196)*x^2)*P(n,x) = (5832+(6804*n+20412)*x+(882*n^2+7938*n+14994)*x^2+(343*n^2+2058*n+3087)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1008*x + 441*x^2.","mathematica":"P[0, x_] := 324 + 1008*x + 441*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((5832 + (6804*n + 20412)*x + (882*n^2 + 7938*n + 14994)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*P[n - 1, x])/(324 + (378*n + 630)*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] = 324 + 1008*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((5832 + (6804*n + 20412)*x + (882*n^2 + 7938*n + 14994)*x^2 + (343*n^2 + 2058*n + 3087)*x^3)*v[i - 1])/(324 + (378*n + 630)*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-7dcf11eddf2b9a8a","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-5a47242ff4b92549","e-5b7193c115f175e8"],"text":"A038321 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A038321 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}