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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":"2b74e96936b6039bdb0a892d5ffad538e5f609cbaa4a840295dee493132a4c3f","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-9a3eef8528def021","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4, read by rows.","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,    12,       3;\n   2: 1,    30,      45,        4;\n   3: 1,    68,     270,      136,         5;\n   4: 1,   146,    1218,     1624,       365,          6;\n   5: 1,   304,    4749,    12992,      7915,        912,          7;\n   6: 1,   622,   16983,    83628,    104535,      33966,       2177,         8;\n   7: 1,  1260,   57480,   470376,   1045350,     705564,     134120,      5040,         9;\n   8: 1,  2538,  187560,  2417984,   8754570,   10505484,    4231472,    500160,     11421,      10;\n   9: 1,  5096,  596943, 11672576,  64930210,  126065808,   90902294,  23345152,   1790829,   25480,    11;\n  10: 1, 10214, 1867269, 53853620, 441376810, 1301808612, 1518776714, 706202896, 121170645, 6224230, 56177, 12;\n","recurrence_sentence":"The row polynomials satisfy P(n,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 >= 3, with P(1,x) = 1 + 12*x + 3*x^2, P(2,x) = 1 + 30*x + 45*x^2 + 4*x^3.","mathematica":"P[1, x_] := 1 + 12*x + 3*x^2;\nP[2, x_] := 1 + 30*x + 45*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-(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 + 12*x + 3*x^2; v[2] = 1 + 30*x + 45*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = -(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-9a3eef8528def021","steps":["\u03b8+1"],"edges":["e-f8fab80111855d04"],"text":"A166344 \u2192 \u03b8+1"},"headline":"A166344 \u2192 \u03b8+1","chain":null}