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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":"9aa66b4671f1bb0c96b7a6a43b9e171397d3d01e4f1573566f572d0ca1cd0c5a","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-8a857e84342335b8","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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,            1020,              675;\n  1:           4913,           26010,            34425,             13500;\n  2:          83521,          589560,          1170450,            918000,            253125;\n  3:        1419857,        12528150,         33162750,          39015000,          21515625,           4556250;\n  4:       24137569,       255574260,        845650125,        1326510000,        1097296875,         464737500,          79734375;\n  5:      410338673,      5068889490,      20126472975,       39463672500,       43526109375,       27651881250,        9488390625,        1366875000;\n  6:     6975757441,     98481281520,     456200054100,     1073411892000,     1479887718750,     1253551950000,      645210562500,      185895000000,       23066015625;\n  7:   118587876497,   1883454509070,    9971229753900,    27372003246000,    45284564193750,    47948362087500,    32905738687500,    14220967500000,     3529100390625,     384433593750;\n  8:  2015993900449,  35576362949100,  211888632270375,   664748650260000,  1283062652156250,  1630244310975000,  1398493894218750,   805854825000000,   299973533203125,   65353710937500,    6343154296875;\n  9: 34271896307633, 665277987148170, 4402574914951125, 15538499699827500, 34276102279031250, 50809281025387500, 52303671643781250, 37673713068750000, 18698350236328125, 6110571972656250, 1186169853515625, 103797070312500;\n","recurrence_sentence":"The row polynomials satisfy (17+(15*n+30)*x)*P(n,x) = (289+(255*n+1020)*x+(225*n+675)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1020*x + 675*x^2.","mathematica":"P[0, x_] := 289 + 1020*x + 675*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((289 + (255*n + 1020)*x + (225*n + 675)*x^2)*P[n - 1, x])/(17 + (15*n + 30)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 1020*x + 675*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((289 + (255*n + 1020)*x + (225*n + 675)*x^2)*v[i - 1])/(17 + (15*n + 30)*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-a5302200d285579d","steps":["basis A013610","\u03b8+1"],"edges":["e-633275fbd6b7b077","e-69894b9a61104828"],"text":"A038331 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038331 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}