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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":"9874f5cd0f502c3ee3c756b2d1d958a561928c417aa4ad2569b045f571902f8d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b2e2c5159c22bcc9","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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:    3,     16;\n  2:    8,     72,      96;\n  3:   21,    264,     720,      512;\n  4:   55,    888,    3648,     5376,     2560;\n  5:  144,   2840,   15552,    35072,    34560,    12288;\n  6:  377,   8784,   60048,   182784,   276480,   202752,    57344;\n  7:  987,  26528,  217296,   833536,  1708800,  1923072,  1118208,   262144;\n  8: 2584,  78696,  751008,  3476736,  9017600, 13768704, 12271616,  5898240,  1179648;\n  9: 6765, 230232, 2507904, 13602048, 42727680, 82667520, 99950592, 73531392, 30081024, 5242880;\n","recurrence_sentence":"The row polynomials satisfy -(80*n-200+(1216*n-1600)*x-(128*n^3-960*n^2-2624*n+3456)*x^2-(1024*n^3-3072*n^2+2048*n)*x^3)*P(n,x) = -(120*n-480+(80*n^2+1744*n-4896)*x-(192*n^3-2368*n^2-7040*n+15360)*x^2-(1792*n^3-7424*n^2-8704*n+14336)*x^3-(2048*n^3-4096*n^2-2048*n+4096)*x^4)*P(n-1,x) - (25+680*x-(80*n^2-320*n-5328)*x^2-(1024*n^2-2176*n-15360)*x^3-(3328*n^2-4096*n-14336)*x^4-(2048*n^2-2048*n-4096)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(120*n - 480 + (80*n^2 + 1744*n - 4896)*x - (192*n^3 - 2368*n^2 - 7040*n + 15360)*x^2 - (1792*n^3 - 7424*n^2 - 8704*n + 14336)*x^3 - (2048*n^3 - 4096*n^2 - 2048*n + 4096)*x^4)*P[n - 1, x] - (25 + 680*x - (80*n^2 - 320*n - 5328)*x^2 - (1024*n^2 - 2176*n - 15360)*x^3 - (3328*n^2 - 4096*n - 14336)*x^4 - (2048*n^2 - 2048*n - 4096)*x^5)*D[P[n - 1, x], x])/(-80*n + 200 - (1216*n - 1600)*x + (128*n^3 - 960*n^2 - 2624*n + 3456)*x^2 + (1024*n^3 - 3072*n^2 + 2048*n)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(120*n - 480 + (80*n^2 + 1744*n - 4896)*x - (192*n^3 - 2368*n^2 - 7040*n + 15360)*x^2 - (1792*n^3 - 7424*n^2 - 8704*n + 14336)*x^3 - (2048*n^3 - 4096*n^2 - 2048*n + 4096)*x^4)*v[i - 1] - (25 + 680*x - (80*n^2 - 320*n - 5328)*x^2 - (1024*n^2 - 2176*n - 15360)*x^3 - (3328*n^2 - 4096*n - 14336)*x^4 - (2048*n^2 - 2048*n - 4096)*x^5)*deriv(v[i - 1], x))/(-80*n + 200 - (1216*n - 1600)*x + (128*n^3 - 960*n^2 - 2624*n + 3456)*x^2 + (1024*n^3 - 3072*n^2 + 2048*n)*x^3)); 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-353fb5a029e4e669","steps":["basis A013611","\u03b8+1"],"edges":["e-a3f64a2b3fc7c9ce","e-fc35585408933284"],"text":"A207607 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A207607 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}