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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":"047a33d315c968f521d11923f050100a333d034cf06bf497fb43668f157695ea","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-1f0d56c568ab8221","role":"sequence","name":"basis_map_A007318 of 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:      3,       6;\n  2:     12,      42,       27;\n  3:     45,     234,      297,      108;\n  4:    171,    1170,     2214,     1620,      405;\n  5:    648,    5508,    13851,    15228,     7695,     1458;\n  6:   2457,   24948,    78246,   114804,    87480,    33534,     5103;\n  7:   9315,  109998,   413343,   758484,   773145,   447606,   137781,   17496;\n  8:  35316,  475470,  2082024,  4586220,  5856705,  4541670,  2112642,  542376,   59049;\n  9: 133893, 2024190, 10120707, 26015580, 39932190, 38835288, 24213735, 9395352, 2066715, 196830;\n","recurrence_sentence":"The row polynomials satisfy (14*n+(7*n^2+19*n)*x+(n^3+6*n^2+5*n)*x^2)*P(n,x) = -(28*n-70+(21*n^2-25*n-158)*x+(2*n^3+7*n^2-83*n-118)*x^2-(6*n^2+36*n+30)*x^3)*P(n-1,x) + (49+(49*n+154)*x+(7*n^2+112*n+176)*x^2+(10*n^2+81*n+86)*x^3+(3*n^2+18*n+15)*x^4)*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[(-(28*n - 70 + (21*n^2 - 25*n - 158)*x + (2*n^3 + 7*n^2 - 83*n - 118)*x^2 - (6*n^2 + 36*n + 30)*x^3)*P[n - 1, x] + (49 + (49*n + 154)*x + (7*n^2 + 112*n + 176)*x^2 + (10*n^2 + 81*n + 86)*x^3 + (3*n^2 + 18*n + 15)*x^4)*D[P[n - 1, x], x])/(14*n + (7*n^2 + 19*n)*x + (n^3 + 6*n^2 + 5*n)*x^2)]];\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] = (-(28*n - 70 + (21*n^2 - 25*n - 158)*x + (2*n^3 + 7*n^2 - 83*n - 118)*x^2 - (6*n^2 + 36*n + 30)*x^3)*v[i - 1] + (49 + (49*n + 154)*x + (7*n^2 + 112*n + 176)*x^2 + (10*n^2 + 81*n + 86)*x^3 + (3*n^2 + 18*n + 15)*x^4)*deriv(v[i - 1], x))/(14*n + (7*n^2 + 19*n)*x + (n^3 + 6*n^2 + 5*n)*x^2)); 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":["x \u21a6 x+1","\u03b8+1"],"edges":["e-30b831f46209b031","e-585c315a2bacc2c4"],"text":"A099093 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A099093 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}