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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":"e6131c80a4c6bb5fa640dd6bb60d64b1031747ee4faf1b7e40b4c04191d52d29","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-392f49b496f79fbf","role":"sequence","name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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: 0, 4;\n   3: 0, 4,  9;\n   4: 0, 0, 18, 16;\n   5: 0, 0,  9, 48,  25;\n   6: 0, 0,  0, 48, 100,  36;\n   7: 0, 0,  0, 16, 150, 180,  49;\n   8: 0, 0,  0,  0, 100, 360, 294,   64;\n   9: 0, 0,  0,  0,  25, 360, 735,  448,  81;\n  10: 0, 0,  0,  0,   0, 180, 980, 1344, 648, 100;\n","recurrence_sentence":"The row polynomials satisfy (n^3-n-(n^2-2*n+1)*x)*P(n,x) = -(2*n^3-2*n^2-4*n-(5*n^2+n+2)*x+2*n*x^2)*P(n-1,x) + ((4*n^2+4*n)*x+(n^2-3*n)*x^2-n*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 2*n^2 - 4*n - (5*n^2 + n + 2)*x + 2*n*x^2)*P[n - 1, x] + ((4*n^2 + 4*n)*x + (n^2 - 3*n)*x^2 - n*x^3)*D[P[n - 1, x], x])/(n^3 - n - (n^2 - 2*n + 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^3 - 2*n^2 - 4*n - (5*n^2 + n + 2)*x + 2*n*x^2)*v[i - 1] + ((4*n^2 + 4*n)*x + (n^2 - 3*n)*x^2 - n*x^3)*deriv(v[i - 1], x))/(n^3 - n - (n^2 - 2*n + 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-392f49b496f79fbf","steps":["(\u03b8+1)\u00b2"],"edges":["e-b1cbb3b54685aec5"],"text":"A026729 \u2192 (\u03b8+1)\u00b2"},"headline":"A026729 \u2192 (\u03b8+1)\u00b2","chain":null}