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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":"9d6d52e60c1f41f37ec57252549d31da4d9f6f2302a34945fdb690864d51e9a9","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-f5a001953ee28e8f","role":"sequence","name":"Riordan array (1, 2+2x).","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: 0, 8;\n  2: 0, 8, 36;\n  3: 0, 0, 72, 128;\n  4: 0, 0, 36, 384,  400;\n  5: 0, 0,  0, 384, 1600,  1152;\n  6: 0, 0,  0, 128, 2400,  5760,  3136;\n  7: 0, 0,  0,   0, 1600, 11520, 18816,   8192;\n  8: 0, 0,  0,   0,  400, 11520, 47040,  57344,  20736;\n  9: 0, 0,  0,   0,    0,  5760, 62720, 172032, 165888, 51200;\n","recurrence_sentence":"The row polynomials satisfy (n^3+3*n^2+2*n-2*n^2*x)*P(n,x) = -(2*n^3+4*n^2-2*n-4-(10*n^2+22*n+16)*x+(8*n+8)*x^2)*P(n-1,x) + ((4*n^2+12*n+8)*x+(2*n^2-2*n-4)*x^2-(4*n+4)*x^3)*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[(-(2*n^3 + 4*n^2 - 2*n - 4 - (10*n^2 + 22*n + 16)*x + (8*n + 8)*x^2)*P[n - 1, x] + ((4*n^2 + 12*n + 8)*x + (2*n^2 - 2*n - 4)*x^2 - (4*n + 4)*x^3)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n - 2*n^2*x)]];\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] = (-(2*n^3 + 4*n^2 - 2*n - 4 - (10*n^2 + 22*n + 16)*x + (8*n + 8)*x^2)*v[i - 1] + ((4*n^2 + 12*n + 8)*x + (2*n^2 - 2*n - 4)*x^2 - (4*n + 4)*x^3)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n - 2*n^2*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-f5a001953ee28e8f","steps":["(\u03b8+1)\u00b2"],"edges":["e-974408a4688a73a4"],"text":"A099040 \u2192 (\u03b8+1)\u00b2"},"headline":"A099040 \u2192 (\u03b8+1)\u00b2","chain":null}