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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":"c982532197258ff89fe5f1a6143951be47a8e6ac58000ebd6b0d744c39b18ed9","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-98aa86c8bf356480","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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:    13,      66,       54;\n  1:    34,     264,      459,       216;\n  2:    89,     930,     2511,      2484,       810;\n  3:   233,    3060,    11259,     17280,     11745,      2916;\n  4:   610,    9648,    45036,     94392,     99225,     51030,     10206;\n  5:  1597,   29544,   167265,    446040,    639900,    507384,    209223,     34992;\n  6:  4181,   88566,   589707,   1912788,   3493125,   3774762,   2393307,    822312,    118098;\n  7: 10946,  261240,  2000403,   7651152,  17012430,  23392152,  20182365,  10637568,   3129597,    393660;\n  8: 28657,  760830,  6587082,  29042280,  76235985, 127646442, 140026320, 100287072,  45172485,  11612970,  1299078;\n  9: 75025, 2193192, 21184578, 105824016, 320604075, 633996720, 846659142, 770313888, 470856726, 185020200, 42220035, 4251528;\n","recurrence_sentence":"The row polynomials satisfy -(60*n+(594*n+810)*x-(54*n^3-1296*n-1782)*x^2-(324*n^3+1458*n^2+2106*n+972)*x^3)*P(n,x) = -(90*n-60+(45*n^2+1071*n+1206)*x-(81*n^3-324*n^2-4941*n-7182)*x^2-(567*n^3+2349*n^2-162*n-5184)*x^3-(486*n^3+2916*n^2+5346*n+2916)*x^4)*P(n-1,x) - (25+435*x-(45*n^2+45*n-2187)*x^2-(432*n^2+1242*n-3051)*x^3-(1053*n^2+3969*n+1296)*x^4-(486*n^2+1944*n+1458)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 13 + 66*x + 54*x^2.","mathematica":"P[0, x_] := 13 + 66*x + 54*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(90*n - 60 + (45*n^2 + 1071*n + 1206)*x - (81*n^3 - 324*n^2 - 4941*n - 7182)*x^2 - (567*n^3 + 2349*n^2 - 162*n - 5184)*x^3 - (486*n^3 + 2916*n^2 + 5346*n + 2916)*x^4)*P[n - 1, x] - (25 + 435*x - (45*n^2 + 45*n - 2187)*x^2 - (432*n^2 + 1242*n - 3051)*x^3 - (1053*n^2 + 3969*n + 1296)*x^4 - (486*n^2 + 1944*n + 1458)*x^5)*D[P[n - 1, x], x])/(-60*n - (594*n + 810)*x + (54*n^3 - 1296*n - 1782)*x^2 + (324*n^3 + 1458*n^2 + 2106*n + 972)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 13 + 66*x + 54*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(90*n - 60 + (45*n^2 + 1071*n + 1206)*x - (81*n^3 - 324*n^2 - 4941*n - 7182)*x^2 - (567*n^3 + 2349*n^2 - 162*n - 5184)*x^3 - (486*n^3 + 2916*n^2 + 5346*n + 2916)*x^4)*v[i - 1] - (25 + 435*x - (45*n^2 + 45*n - 2187)*x^2 - (432*n^2 + 1242*n - 3051)*x^3 - (1053*n^2 + 3969*n + 1296)*x^4 - (486*n^2 + 1944*n + 1458)*x^5)*deriv(v[i - 1], x))/(-60*n - (594*n + 810)*x + (54*n^3 - 1296*n - 1782)*x^2 + (324*n^3 + 1458*n^2 + 2106*n + 972)*x^3)); 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-dfab35176bd08e72","steps":["basis A013610","\u03b8+1"],"edges":["e-77ee9774535f66a4","e-451e7df0c2fbf35c"],"text":"A207606 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A207606 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}