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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":"94dea0cb665efa82b6677a8b0b61754569c301e7c820d8c0401863e0440eb632","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-aac530d688cc3150","role":"sequence","name":"Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","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: 1,        2;\n  2: 1,       34,          3;\n  3: 1,      236,        354,           4;\n  4: 1,     1458,       8043,        2916,            5;\n  5: 1,     8800,     125235,      166980,        22000,            6;\n  6: 1,    52862,    1665720,     5396980,      2776200,       158586,           7;\n  7: 1,   317244,   20450538,   133826740,    167283425,     40901076,     1110354,          8;\n  8: 1,  1903546,  240104229,  2861284624,   7059634975,   4291926936,   560243201,    7614184,        9;\n  9: 1, 11421368, 2746317333, 55841199036, 242080501880, 290496602256, 97722098313, 7323512888, 51396156, 10;\n","recurrence_sentence":"The row polynomials satisfy (20*n-21)*P(n,x) = -(500*n-25)*x^2*P(n-4,x) + ((800*n-40)*x-(1000*n^2-1850*n-35)*x^2)*P(n-3,x) - ((500*n-25)*x^2-(500*n-25)*x^3)*d/dx P(n-3,x) + (80*n-4+(800*n^2-1280*n-138)*x-(500*n^3-1325*n^2+660*n+189)*x^2)*P(n-2,x) + ((400*n-20)*x-(500*n^2-25*n-125)*x^2+(500*n^2-425*n-105)*x^3)*d/dx P(n-2,x) - (60*n+17-(200*n^2-370*n+168)*x)*P(n-1,x) + ((100*n-105)*x-(100*n-105)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 34*x + 3*x^2, P(3,x) = 1 + 236*x + 354*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 1 + 34*x + 3*x^2;\nP[3, x_] := 1 + 236*x + 354*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(500*n - 25)*x^2*P[n - 4, x] + ((800*n - 40)*x - (1000*n^2 - 1850*n - 35)*x^2)*P[n - 3, x] - ((500*n - 25)*x^2 - (500*n - 25)*x^3)*D[P[n - 3, x], x] + (80*n - 4 + (800*n^2 - 1280*n - 138)*x - (500*n^3 - 1325*n^2 + 660*n + 189)*x^2)*P[n - 2, x] + ((400*n - 20)*x - (500*n^2 - 25*n - 125)*x^2 + (500*n^2 - 425*n - 105)*x^3)*D[P[n - 2, x], x] - (60*n + 17 - (200*n^2 - 370*n + 168)*x)*P[n - 1, x] + ((100*n - 105)*x - (100*n - 105)*x^2)*D[P[n - 1, x], x])/(20*n - 21)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 34*x + 3*x^2; v[4] = 1 + 236*x + 354*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(500*n - 25)*x^2*v[i - 4] + ((800*n - 40)*x - (1000*n^2 - 1850*n - 35)*x^2)*v[i - 3] - ((500*n - 25)*x^2 - (500*n - 25)*x^3)*deriv(v[i - 3], x) + (80*n - 4 + (800*n^2 - 1280*n - 138)*x - (500*n^3 - 1325*n^2 + 660*n + 189)*x^2)*v[i - 2] + ((400*n - 20)*x - (500*n^2 - 25*n - 125)*x^2 + (500*n^2 - 425*n - 105)*x^3)*deriv(v[i - 2], x) - (60*n + 17 - (200*n^2 - 370*n + 168)*x)*v[i - 1] + ((100*n - 105)*x - (100*n - 105)*x^2)*deriv(v[i - 1], x))/(20*n - 21)); 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-aac530d688cc3150","steps":["\u03b8+1"],"edges":["e-fe3642eed5cabd2c"],"text":"A144442 \u2192 \u03b8+1"},"headline":"A144442 \u2192 \u03b8+1","chain":null}