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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":"ba8ff6cc150f626dee86147df4887c988cc837a26eda1ec63e1efe7338cd5766","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-65e47d0258ed4170","role":"sequence","name":"Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).","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,       20,           3;\n  3: 1,      182,         105,            4;\n  4: 1,     1640,        2898,          336,           5;\n  5: 1,    14762,       74910,        20328,         825,          6;\n  6: 1,   132860,     1894893,      1095952,       92235,       1716,         7;\n  7: 1,  1195742,    47571615,     56228172,     8840975,     318318,      3185,       8;\n  8: 1, 10761680,  1191083988,   2818609248,   786404190,   49125648,    909636,    5440,    9;\n  9: 1, 96855122, 29793242220, 139699965136, 67222000950, 6887888436, 211041740, 2263584, 8721, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+2*x+x^2)*P(n-2,x) - (8*x+4*x^2)*d/dx P(n-2,x) - (4*x^2+4*x^3)*d^2/dx^2 P(n-2,x) + (2+2*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) + 4*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(1 + 2*x + x^2)*P[n - 2, x] - (8*x + 4*x^2)*D[P[n - 2, x], x] - (4*x^2 + 4*x^3)*D[P[n - 2, x], {x, 2}] + (2 + 2*x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x] + 4*x^2*D[P[n - 1, x], {x, 2}]];\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; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 2*x + x^2)*v[i - 2] - (8*x + 4*x^2)*deriv(v[i - 2], x) - (4*x^2 + 4*x^3)*deriv(deriv(v[i - 2], x), x) + (2 + 2*x)*v[i - 1] + 8*x*deriv(v[i - 1], x) + 4*x^2*deriv(deriv(v[i - 1], x), x)); 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-65e47d0258ed4170","steps":["\u03b8+1"],"edges":["e-00d68563ae24f541"],"text":"A160562 \u2192 \u03b8+1"},"headline":"A160562 \u2192 \u03b8+1","chain":null}