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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":"eb01bfd00a161d9b0f167ce6061b77673a3176837d94d3784198427b4068542a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-09153823987c7e65","role":"sequence","name":"Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1,    14,       3;\n   2: 1,    42,      63,         4;\n   3: 1,   102,     483,       204,          5;\n   4: 1,   226,    2439,      3252,        565,          6;\n   5: 1,   478,   10083,     30524,      16805,       1434,          7;\n   6: 1,   986,   37263,    215332,     269165,      74526,       3451,          8;\n   7: 1,  2006,  128595,   1280428,    3035045,    1920642,     300055,       8024,         9;\n   8: 1,  4050,  424647,   6818772,   27470085,   32964102,   11932851,    1132392,     18225,       10;\n   9: 1,  8142, 1361715,  33690716,  214620565,  435631818,  300468791,   67381432,   4085145,    40710,    11;\n  10: 1, 16330, 4280175, 157897796, 1512291565, 4812172878, 5614201691, 2419666504, 355270041, 14267250, 89815, 12;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+2*n)*x^2*P(n-4,x) - (2*n^2+4*n)*x^2*P(n-3,x) - ((n+2)*x^2-(n+2)*x^3)*d/dx P(n-3,x) + ((2*n+4)*x-(n^2+2*n)*x^2)*P(n-2,x) - ((n+2)*x^2-(n+2)*x^3)*d/dx P(n-2,x) + (1+(2*n+2)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1 + 14*x + 3*x^2, P(2,x) = 1 + 42*x + 63*x^2 + 4*x^3, P(3,x) = 1 + 102*x + 483*x^2 + 204*x^3 + 5*x^4, P(4,x) = 1 + 226*x + 2439*x^2 + 3252*x^3 + 565*x^4 + 6*x^5.","mathematica":"P[1, x_] := 1 + 14*x + 3*x^2;\nP[2, x_] := 1 + 42*x + 63*x^2 + 4*x^3;\nP[3, x_] := 1 + 102*x + 483*x^2 + 204*x^3 + 5*x^4;\nP[4, x_] := 1 + 226*x + 2439*x^2 + 3252*x^3 + 565*x^4 + 6*x^5;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n)*x^2*P[n - 4, x] - (2*n^2 + 4*n)*x^2*P[n - 3, x] - ((n + 2)*x^2 - (n + 2)*x^3)*D[P[n - 3, x], x] + ((2*n + 4)*x - (n^2 + 2*n)*x^2)*P[n - 2, x] - ((n + 2)*x^2 - (n + 2)*x^3)*D[P[n - 2, x], x] + (1 + (2*n + 2)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 14*x + 3*x^2; v[2] = 1 + 42*x + 63*x^2 + 4*x^3; v[3] = 1 + 102*x + 483*x^2 + 204*x^3 + 5*x^4; v[4] = 1 + 226*x + 2439*x^2 + 3252*x^3 + 565*x^4 + 6*x^5; for(n = 5, N, my(i = n); v[i] = -(n^2 + 2*n)*x^2*v[i - 4] - (2*n^2 + 4*n)*x^2*v[i - 3] - ((n + 2)*x^2 - (n + 2)*x^3)*deriv(v[i - 3], x) + ((2*n + 4)*x - (n^2 + 2*n)*x^2)*v[i - 2] - ((n + 2)*x^2 - (n + 2)*x^3)*deriv(v[i - 2], x) + (1 + (2*n + 2)*x)*v[i - 1] + (x - x^2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-09153823987c7e65","steps":["\u03b8+1"],"edges":["e-afceae6d3b9db17c"],"text":"A046739 \u2192 \u03b8+1"},"headline":"A046739 \u2192 \u03b8+1","chain":null}