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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":"77002f5e6a4bb36d9905669a73b67afc6ccf812b396f99b42411356a6c0764e6","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-0778fc04c5956f01","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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:       5,       12;\n  2:      24,      120,        81;\n  3:     115,      876,      1215,       432;\n  4:     551,     5640,     11907,      8640,      2025;\n  5:    2640,    33936,     96390,    106272,     50625,      8748;\n  6:   12649,   195720,    699111,   1036800,    749250,    262440,     35721;\n  7:   60605,  1096452,   4720275,   8806320,   8555625,   4540212,   1250235,    139968;\n  8:  290376,  6013800,  30303315,  68169600,  83308500,  59398920,  24754653,   5598720,   531441;\n  9: 1391275, 32457060, 187389450, 493659360, 727531875, 652347108, 365068620, 124851456, 23914845, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (392*n+70*n*x-(420*n^3-50*n)*x^2-150*n^3*x^3+18*n^5*x^4)*P(n,x) = (980*n+980+(1029*n^2+2527*n+1498)*x-(1050*n^3+1155*n^2-545*n-650)*x^2-(126*n^4+1446*n^3+2391*n^2+1149*n+78)*x^3+(45*n^5+45*n^4-405*n^3-810*n^2-495*n-90)*x^4+(27*n^5+81*n^4+81*n^3+27*n^2)*x^5)*P(n-1,x) + (343+735*x-(1029*n^2+1029*n-378)*x^2-(1995*n^2+1995*n+170)*x^3+(63*n^4+126*n^3-1128*n^2-1191*n-201)*x^4+(90*n^4+180*n^3-135*n^2-225*n-45)*x^5+(27*n^4+54*n^3+27*n^2)*x^6)*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[((980*n + 980 + (1029*n^2 + 2527*n + 1498)*x - (1050*n^3 + 1155*n^2 - 545*n - 650)*x^2 - (126*n^4 + 1446*n^3 + 2391*n^2 + 1149*n + 78)*x^3 + (45*n^5 + 45*n^4 - 405*n^3 - 810*n^2 - 495*n - 90)*x^4 + (27*n^5 + 81*n^4 + 81*n^3 + 27*n^2)*x^5)*P[n - 1, x] + (343 + 735*x - (1029*n^2 + 1029*n - 378)*x^2 - (1995*n^2 + 1995*n + 170)*x^3 + (63*n^4 + 126*n^3 - 1128*n^2 - 1191*n - 201)*x^4 + (90*n^4 + 180*n^3 - 135*n^2 - 225*n - 45)*x^5 + (27*n^4 + 54*n^3 + 27*n^2)*x^6)*D[P[n - 1, x], x])/(392*n + 70*n*x - (420*n^3 - 50*n)*x^2 - 150*n^3*x^3 + 18*n^5*x^4)]];\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] = ((980*n + 980 + (1029*n^2 + 2527*n + 1498)*x - (1050*n^3 + 1155*n^2 - 545*n - 650)*x^2 - (126*n^4 + 1446*n^3 + 2391*n^2 + 1149*n + 78)*x^3 + (45*n^5 + 45*n^4 - 405*n^3 - 810*n^2 - 495*n - 90)*x^4 + (27*n^5 + 81*n^4 + 81*n^3 + 27*n^2)*x^5)*v[i - 1] + (343 + 735*x - (1029*n^2 + 1029*n - 378)*x^2 - (1995*n^2 + 1995*n + 170)*x^3 + (63*n^4 + 126*n^3 - 1128*n^2 - 1191*n - 201)*x^4 + (90*n^4 + 180*n^3 - 135*n^2 - 225*n - 45)*x^5 + (27*n^4 + 54*n^3 + 27*n^2)*x^6)*deriv(v[i - 1], x))/(392*n + 70*n*x - (420*n^3 - 50*n)*x^2 - 150*n^3*x^3 + 18*n^5*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-30cb3cad216ed189","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-99fce6e1b50f281f","e-13beba469b000f33"],"text":"A207823 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A207823 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}