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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":"298a48f94fa61aa16da14640f7ea1f910bf4edba4c26986ca3cff98b317c3e7a","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-9e68379063b83f9d","role":"sequence","name":"Triangle read by rows; T(n,k) is the number of achiral rows of n colors using exactly k colors.","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;\n  2: 0, 4;\n  3: 0, 4,  18;\n  4: 0, 4,  18;\n  5: 0, 4,  54,   96;\n  6: 0, 4,  54,   96;\n  7: 0, 4, 126,  576,  600;\n  8: 0, 4, 126,  576,  600;\n  9: 0, 4, 270, 2400, 6000, 4320;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*x-2*x^2)*P(n-4,x) + (x-x^3)*d/dx P(n-4,x) - (1-4*x)*P(n-2,x) + (x+x^2)*d/dx P(n-2,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4*x, P(2,x) = 4*x, P(3,x) = 4*x + 18*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x;\nP[2, x_] := 4*x;\nP[3, x_] := 4*x + 18*x^2;\nP[n_, x_] := P[n, x] = Expand[(4*x - 2*x^2)*P[n - 4, x] + (x - x^3)*D[P[n - 4, x], x] - (1 - 4*x)*P[n - 2, x] + (x + x^2)*D[P[n - 2, 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; v[3] = 4*x; v[4] = 4*x + 18*x^2; for(n = 4, N, my(i = n + 1); v[i] = (4*x - 2*x^2)*v[i - 4] + (x - x^3)*deriv(v[i - 4], x) - (1 - 4*x)*v[i - 2] + (x + x^2)*deriv(v[i - 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-9e68379063b83f9d","steps":["(\u03b8+1)\u00b2"],"edges":["e-7b30e2a5292945d7"],"text":"A321434 \u2192 (\u03b8+1)\u00b2"},"headline":"A321434 \u2192 (\u03b8+1)\u00b2","chain":null}