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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":"e1349f9ee8f3d97c25fb5756c8d051af2818d5af73b4fff0c1417663839fc9dc","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-b8808e9caf8dcf35","role":"sequence","name":"Triangle read by rows, (2, 3)-Lah numbers.","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:                 36,                  4;\n  2:               1764,                400,                  9;\n  3:             112896,              39456,               1800,               16;\n  4:            9144576,            4397184,             306576,             5504,              25;\n  5:          914457600,          568637568,           54040608,          1475200,           13500,            36;\n  6:       110649369600,        85541640192,        10412297280,        385211392,         5342500,         28656,          49;\n  7:     15933509222400,     14899134670848,      2233278957312,     105183313920,      1969572800,      15974784,       54880,       64;\n  8:   2692763058585600,   2983964432375808,    536010818404608,   30897202065408,    733555467200,    8012014848,    41555136,    97280,     81;\n  9: 527781559482777600, 682163049518899200, 143932689484425216, 9882199518511104, 285948849600000, 3948657232896, 27527296832, 97176576, 162324, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4+14*n^3+73*n^2+168*n+144+(4*n^2+32*n+60)*x-9*x^2)*P(n-2,x) - ((4*n^3+47*n^2+182*n+231)*x-(4*n^2+32*n+69)*x^2-4*x^3)*d/dx P(n-2,x) - ((5*n^2+40*n+75)*x^2-(8*n+41)*x^3)*d^2/dx^2 P(n-2,x) - ((2*n+6)*x^3-4*x^4)*d^3/dx^3 P(n-2,x) + (2*n^2+16*n+34+4*x)*P(n-1,x) + ((4*n+17)*x-4*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 36 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 36 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 + 14*n^3 + 73*n^2 + 168*n + 144 + (4*n^2 + 32*n + 60)*x - 9*x^2)*P[n - 2, x] - ((4*n^3 + 47*n^2 + 182*n + 231)*x - (4*n^2 + 32*n + 69)*x^2 - 4*x^3)*D[P[n - 2, x], x] - ((5*n^2 + 40*n + 75)*x^2 - (8*n + 41)*x^3)*D[P[n - 2, x], {x, 2}] - ((2*n + 6)*x^3 - 4*x^4)*D[P[n - 2, x], {x, 3}] + (2*n^2 + 16*n + 34 + 4*x)*P[n - 1, x] + ((4*n + 17)*x - 4*x^2)*D[P[n - 1, x], x] + 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] = 36 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 14*n^3 + 73*n^2 + 168*n + 144 + (4*n^2 + 32*n + 60)*x - 9*x^2)*v[i - 2] - ((4*n^3 + 47*n^2 + 182*n + 231)*x - (4*n^2 + 32*n + 69)*x^2 - 4*x^3)*deriv(v[i - 2], x) - ((5*n^2 + 40*n + 75)*x^2 - (8*n + 41)*x^3)*deriv(deriv(v[i - 2], x), x) - ((2*n + 6)*x^3 - 4*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2*n^2 + 16*n + 34 + 4*x)*v[i - 1] + ((4*n + 17)*x - 4*x^2)*deriv(v[i - 1], x) + 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-b8808e9caf8dcf35","steps":["(\u03b8+1)\u00b2"],"edges":["e-bab6190e354efb1f"],"text":"A371259 \u2192 (\u03b8+1)\u00b2"},"headline":"A371259 \u2192 (\u03b8+1)\u00b2","chain":null}