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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":"d377b7fab6f832a6a480d335fe2827f4a0951593eb3ea57394de09897bdd8606","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-f0f7aa17c45804ab","role":"sequence","name":"veronese_section_2_0 of Triangle of coefficients in expansion of (1+4x)^n.","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;\n  2: 1,   32;\n  3: 1,   96;\n  4: 1,  192,   768;\n  5: 1,  320,  3840;\n  6: 1,  480, 11520,   16384;\n  7: 1,  672, 26880,  114688;\n  8: 1,  896, 53760,  458752,  327680;\n  9: 1, 1152, 96768, 1376256, 2949120;\n","recurrence_sentence":"The row polynomials satisfy -(3*n-3-(8*n^3-8*n)*x)*P(n,x) = -(3*n-3-(8*n^3-104*n+96)*x-(128*n^3+256*n^2-128*n-256)*x^2)*P(n-1,x) - (6*x-(16*n^2+48*n+128)*x^2+(256*n^2+768*n+512)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 - (8*n^3 - 104*n + 96)*x - (128*n^3 + 256*n^2 - 128*n - 256)*x^2)*P[n - 1, x] - (6*x - (16*n^2 + 48*n + 128)*x^2 + (256*n^2 + 768*n + 512)*x^3)*D[P[n - 1, x], x])/(-3*n + 3 + (8*n^3 - 8*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n - 3 - (8*n^3 - 104*n + 96)*x - (128*n^3 + 256*n^2 - 128*n - 256)*x^2)*v[i - 1] - (6*x - (16*n^2 + 48*n + 128)*x^2 + (256*n^2 + 768*n + 512)*x^3)*deriv(v[i - 1], x))/(-3*n + 3 + (8*n^3 - 8*n)*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-2e0409c0014ec240","steps":["Veronese 2","\u03b8+1"],"edges":["e-d16c364b5af70132","e-d0933ddcd270f683"],"text":"A013611 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A013611 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}