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(f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. 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":"77fc28c3d94df4118beac8fa5f46568b7a9d4e346ed5f104388f6d101995e7c3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5fa21c8779d2bdd1","role":"sequence","name":"Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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: 1,  12;\n  2: 1,  28,    27;\n  3: 1,  48,   198,     48;\n  4: 1,  72,   621,    640,      75;\n  5: 1, 100,  1431,   3856,    1525,     108;\n  6: 1, 132,  2790,  14128,   14550,    3060,     147;\n  7: 1, 168,  4887,  39440,   84825,   41580,    5488,    192;\n  8: 1, 208,  7938,  92768,  345125,  352296,   99568,   9088,   243;\n  9: 1, 252, 12186, 193760, 1106325, 2070756, 1145032, 210816, 14175, 300;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (9*n^3+9*n^2-18*n)*x^2*P(n-4,x) + (4*n^3+4*n^2-8*n)*x^3*d/dx P(n-4,x) - ((4*n^2+8*n)*x-(18*n^2+27*n)*x^2)*P(n-3,x) + ((4*n^2+8*n)*x^2+(8*n^2+12*n)*x^3)*d/dx P(n-3,x) - (n+2-(4*n^2+4*n-4)*x-(9*n+9)*x^2)*P(n-2,x) - ((4*n^2+4*n-4)*x^2-(4*n+4)*x^3)*d/dx P(n-2,x) + (2*n+3+(4*n+4)*x)*P(n-1,x) - (4*n+4)*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 12*x, P(2,x) = 1 + 28*x + 27*x^2, P(3,x) = 1 + 48*x + 198*x^2 + 48*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 12*x;\nP[2, x_] := 1 + 28*x + 27*x^2;\nP[3, x_] := 1 + 48*x + 198*x^2 + 48*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^3 + 9*n^2 - 18*n)*x^2*P[n - 4, x] + (4*n^3 + 4*n^2 - 8*n)*x^3*D[P[n - 4, x], x] - ((4*n^2 + 8*n)*x - (18*n^2 + 27*n)*x^2)*P[n - 3, x] + ((4*n^2 + 8*n)*x^2 + (8*n^2 + 12*n)*x^3)*D[P[n - 3, x], x] - (n + 2 - (4*n^2 + 4*n - 4)*x - (9*n + 9)*x^2)*P[n - 2, x] - ((4*n^2 + 4*n - 4)*x^2 - (4*n + 4)*x^3)*D[P[n - 2, x], x] + (2*n + 3 + (4*n + 4)*x)*P[n - 1, x] - (4*n + 4)*x^2*D[P[n - 1, x], x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 12*x; v[3] = 1 + 28*x + 27*x^2; v[4] = 1 + 48*x + 198*x^2 + 48*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((9*n^3 + 9*n^2 - 18*n)*x^2*v[i - 4] + (4*n^3 + 4*n^2 - 8*n)*x^3*deriv(v[i - 4], x) - ((4*n^2 + 8*n)*x - (18*n^2 + 27*n)*x^2)*v[i - 3] + ((4*n^2 + 8*n)*x^2 + (8*n^2 + 12*n)*x^3)*deriv(v[i - 3], x) - (n + 2 - (4*n^2 + 4*n - 4)*x - (9*n + 9)*x^2)*v[i - 2] - ((4*n^2 + 4*n - 4)*x^2 - (4*n + 4)*x^3)*deriv(v[i - 2], x) + (2*n + 3 + (4*n + 4)*x)*v[i - 1] - (4*n + 4)*x^2*deriv(v[i - 1], x))/(n + 1)); 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-5fa21c8779d2bdd1","steps":["(\u03b8+1)\u00b2"],"edges":["e-7a524eebea10f044"],"text":"A249248 \u2192 (\u03b8+1)\u00b2"},"headline":"A249248 \u2192 (\u03b8+1)\u00b2","chain":null}