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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":"1546178ec872c2ab0b20c0e7c3b155d4b9fb36b89a252029abb495e3d44a36b9","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-016d7724d51f6416","role":"sequence","name":"Riordan array (1, 3+3x).","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, 12;\n  2: 0, 12,  81;\n  3: 0,  0, 162,  432;\n  4: 0,  0,  81, 1296,  2025;\n  5: 0,  0,   0, 1296,  8100,  8748;\n  6: 0,  0,   0,  432, 12150, 43740,  35721;\n  7: 0,  0,   0,    0,  8100, 87480, 214326,  139968;\n  8: 0,  0,   0,    0,  2025, 87480, 535815,  979776,  531441;\n  9: 0,  0,   0,    0,     0, 43740, 714420, 2939328, 4251528, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (n^3+3*n^2+2*n-3*n^2*x)*P(n,x) = -(2*n^3+4*n^2-2*n-4-(15*n^2+33*n+24)*x+(18*n+18)*x^2)*P(n-1,x) + ((4*n^2+12*n+8)*x+(3*n^2-3*n-6)*x^2-(9*n+9)*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 - (15*n^2 + 33*n + 24)*x + (18*n + 18)*x^2)*P[n - 1, x] + ((4*n^2 + 12*n + 8)*x + (3*n^2 - 3*n - 6)*x^2 - (9*n + 9)*x^3)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n - 3*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 - (15*n^2 + 33*n + 24)*x + (18*n + 18)*x^2)*v[i - 1] + ((4*n^2 + 12*n + 8)*x + (3*n^2 - 3*n - 6)*x^2 - (9*n + 9)*x^3)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n - 3*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-016d7724d51f6416","steps":["(\u03b8+1)\u00b2"],"edges":["e-4a001f0a65c0dd59"],"text":"A099093 \u2192 (\u03b8+1)\u00b2"},"headline":"A099093 \u2192 (\u03b8+1)\u00b2","chain":null}