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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":"60716fca1867a80002c4db0bbf336fc9ce85edf26b82430a0c681e9790541922","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-a9d89adf599d215a","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^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:            1;\n  1:           19,            32;\n  2:          361,          1216,            768;\n  3:         6859,         34656,          43776,          16384;\n  4:       130321,        877952,        1663488,        1245184,          327680;\n  5:      2476099,      20851360,       52677120,       59146240,        31129600,         6291456;\n  6:     47045881,     475411008,     1501297920,     2247557120,      1774387200,       717225984,      117440512;\n  7:    893871739,   10538277344,    39934524672,    74731274240,     78664499200,     47695527936,    15619588096,     2147483648;\n  8:  16983563041,  228831165184,  1011674625024,  2271830736896,   2989250969600,   2416573415424,  1187088695296,   326417514496,   38654705664;\n  9: 322687697779, 4891266155808, 24713765839872, 64747176001536, 102232383160320, 103308513509376, 67664055631872, 27908697489408, 6609954668544, 687194767360;\n","recurrence_sentence":"The row polynomials satisfy (19+16*n*x)*P(n,x) = (361+(304*n+608)*x+(256*n+256)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((361 + (304*n + 608)*x + (256*n + 256)*x^2)*P[n - 1, x])/(19 + 16*n*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((361 + (304*n + 608)*x + (256*n + 256)*x^2)*v[i - 1])/(19 + 16*n*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-2f02b2035f8f4541","steps":["basis A013609","\u03b8+1"],"edges":["e-6db2596ebcec1df1","e-0f0a090a7f43305d"],"text":"A038322 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038322 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}