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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":"54b1b49129ce2923c412c72a2604d1561741f75ebad664681452bff61cfda9ca","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-e1c721301bf80375","role":"sequence","name":"Triangle read by rows giving successive states of cellular automaton generated by \"Rule 84\" initiated with a single ON (black) cell.","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, 4, 9;\n  2: 0, 0, 0, 16, 25;\n  3: 0, 0, 0,  0,  0, 36, 49;\n  4: 0, 0, 0,  0,  0,  0,  0, 64, 81;\n  5: 0, 0, 0,  0,  0,  0,  0,  0,  0, 100, 121;\n  6: 0, 0, 0,  0,  0,  0,  0,  0,  0,   0,   0, 144, 169;\n  7: 0, 0, 0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0, 196, 225;\n  8: 0, 0, 0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0, 256, 289;\n  9: 0, 0, 0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,   0,   0, 324, 361;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 7*x^4*P(n-2,x) + 2*x^5*d/dx P(n-2,x) + 6*x^2*P(n-1,x) - 2*x^3*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x + 9*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[7*x^4*P[n - 2, x] + 2*x^5*D[P[n - 2, x], x] + 6*x^2*P[n - 1, x] - 2*x^3*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = 7*x^4*v[i - 2] + 2*x^5*deriv(v[i - 2], x) + 6*x^2*v[i - 1] - 2*x^3*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-e1c721301bf80375","steps":["(\u03b8+1)\u00b2"],"edges":["e-abf16a62d8f33744"],"text":"A267006 \u2192 (\u03b8+1)\u00b2"},"headline":"A267006 \u2192 (\u03b8+1)\u00b2","chain":null}