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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":"07b6a435e3845c90befa4373ad2fb68f5974e4e6f39c93c74ea8acb379bf5f6a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-15df20466f2d1f05","role":"sequence","name":"Stirling-like number triangle defined by sequence A000217.","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,     4;\n  2: 1,    16,       9;\n  3: 1,    52,      90,       16;\n  4: 1,   160,     657,      320,        25;\n  5: 1,   484,    4302,     4368,       875,       36;\n  6: 1,  1456,   26901,    51328,     19950,     2016,      49;\n  7: 1,  4372,  164682,   561104,    379450,    71064,    4116,     64;\n  8: 1, 13120,  997929,  5903808,   6568475,  2038752,  211974,   7680,    81;\n  9: 1, 39364, 6017094, 60812176, 107751825, 52272396, 8710240, 553344, 13365, 100;\n","recurrence_sentence":"The row polynomials satisfy 4*P(n,x) = -(8*x-36*x^2)*P(n-2,x) - (6*x-26*x^2-16*x^3)*d/dx P(n-2,x) - (6*x^2-42*x^3)*d^2/dx^2 P(n-2,x) - (x^3-8*x^4)*d^3/dx^3 P(n-2,x) + (4+16*x)*P(n-1,x) + (10*x-16*x^2)*d/dx P(n-1,x) + 2*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*x - 36*x^2)*P[n - 2, x] - (6*x - 26*x^2 - 16*x^3)*D[P[n - 2, x], x] - (6*x^2 - 42*x^3)*D[P[n - 2, x], {x, 2}] - (x^3 - 8*x^4)*D[P[n - 2, x], {x, 3}] + (4 + 16*x)*P[n - 1, x] + (10*x - 16*x^2)*D[P[n - 1, x], x] + 2*x^2*D[P[n - 1, x], {x, 2}])/(4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*x - 36*x^2)*v[i - 2] - (6*x - 26*x^2 - 16*x^3)*deriv(v[i - 2], x) - (6*x^2 - 42*x^3)*deriv(deriv(v[i - 2], x), x) - (x^3 - 8*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (4 + 16*x)*v[i - 1] + (10*x - 16*x^2)*deriv(v[i - 1], x) + 2*x^2*deriv(deriv(v[i - 1], x), x))/(4)); 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-15df20466f2d1f05","steps":["(\u03b8+1)\u00b2"],"edges":["e-fab3cf82fcd21c6d"],"text":"A080248 \u2192 (\u03b8+1)\u00b2"},"headline":"A080248 \u2192 (\u03b8+1)\u00b2","chain":null}