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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":"9c98468225728cd168e494ce084b1d089bc75f4ff06a9c8df09a4ab9c034a10c","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-adb5eb667a229c4c","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 6).","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:        36,           36,             3;\n  1:       216,          504,           108,             4;\n  2:      1296,         6480,          2700,           240,             5;\n  3:      7776,        80352,         58320,          9360,           450,            6;\n  4:     46656,       979776,       1170288,        302400,         25200,          756,           7;\n  5:    279936,     11850624,      22534848,       8817984,       1134000,        57456,        1176,          8;\n  6:   1679616,    142767360,     423403200,     241678080,      45042480,      3429216,      116424,       1728,        9;\n  7:  10077696,   1716567552,    7835408640,    6364811520,    1653372000,    177502752,     8890560,     216000,     2430,     10;\n  8:  60466176,  20618966016,  143612206848,  163202688000,   57557174400,   8374145472,   580490064,   20528640,   374220,   3300,   11;\n  9: 362797056, 247548524544, 2615948172288, 4108347454464, 1930718592000, 370537846272, 34150419072, 1648791936, 43302600, 613800, 4356, 12;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 6*x^2*d/dx P(n-2,x) + (6+2*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 36 + 36*x + 3*x^2, P(1,x) = 216 + 504*x + 108*x^2 + 4*x^3.","mathematica":"P[0, x_] := 36 + 36*x + 3*x^2;\nP[1, x_] := 216 + 504*x + 108*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 6*x^2*D[P[n - 2, x], x] + (6 + 2*x)*P[n - 1, x] + 6*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 36 + 36*x + 3*x^2; v[2] = 216 + 504*x + 108*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 6*x^2*deriv(v[i - 2], x) + (6 + 2*x)*v[i - 1] + 6*x*deriv(v[i - 1], 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-adb5eb667a229c4c","steps":["\u03b8+1"],"edges":["e-3935bfcee782bee7"],"text":"A075501 \u2192 \u03b8+1"},"headline":"A075501 \u2192 \u03b8+1","chain":null}