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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":"c67b8e481fb41a067c7c5bc0b8e85bf02b60f7c95039548c397764d6a8a37272","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-170093e6e241015d","role":"sequence","name":"Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   1;\n   2:   2,      12;\n   3:   4,      84,        162;\n   4:   8,     468,       2430,        2592;\n   5:  16,    2436,      25758,       67392,        48600;\n   6:  32,   12372,     238950,     1153440,      1944000,       1049760;\n   7:  64,   62244,    2078622,    16511040,     48843000,      59836320,      25719120;\n   8: 128,  311988,   17469270,   214879392,    993384000,    2072226240,    1980372240,     705438720;\n   9: 256, 1561476,  143965998,  2643181632,  17936364600,   56684940480,   90376987680,   70543872000,   21427701120;\n  10: 512, 7810452, 1172807910, 31378453920, 300668760000, 1351069463520, 3196320795360, 4101420718080, 2699890341120, 714256704000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+12*x-18*x^2)*P(n-2,x) + (3*x-6*x^2-9*x^3)*d/dx P(n-2,x) + (1+12*x)*P(n-1,x) + (3*x+3*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 12*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 12*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 12*x - 18*x^2)*P[n - 2, x] + (3*x - 6*x^2 - 9*x^3)*D[P[n - 2, x], x] + (1 + 12*x)*P[n - 1, x] + (3*x + 3*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 12*x; for(n = 3, N, my(i = n); v[i] = (2 + 12*x - 18*x^2)*v[i - 2] + (3*x - 6*x^2 - 9*x^3)*deriv(v[i - 2], x) + (1 + 12*x)*v[i - 1] + (3*x + 3*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-170093e6e241015d","steps":["(\u03b8+1)\u00b2"],"edges":["e-9ff2ec4f4719312a"],"text":"A225472 \u2192 (\u03b8+1)\u00b2"},"headline":"A225472 \u2192 (\u03b8+1)\u00b2","chain":null}