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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":"1b1ddc5a8f77135b873501f12cb885b4202c6f4b9c8b69201c53b74c40ac1e72","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-c77d5d4d44ca1f43","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 2.","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:   2,        8;\n  2:   4,       80,         36;\n  3:   8,      528,       1188,         128;\n  4:  16,     2992,      19008,       11968,          400;\n  5:  32,    15856,     226116,      401984,        99100,         1152;\n  6:  64,    81456,    2308392,     8843648,      6412200,       733104,         3136;\n  7: 128,   412400,   21582828,   154733440,    241771000,     86331312,      5051900,        8192;\n  8: 256,  2073776,  191220624,  2354348864,   6769588000,   5297284944,   1041090064,    33180416,     20736;\n  9: 512, 10395504, 1637082900, 32696793024, 157311623700, 226528738128, 100133928636, 11641478400, 210508956, 51200;\n","recurrence_sentence":"The row polynomials satisfy (9*n^2-6*n+1)*P(n,x) = (18*n^2+6*n-4-(216*n^3-216*n^2-252*n+208)*x+(729*n^4-1134*n^3+81*n^2+216*n-48)*x^2)*P(n-2,x) - ((108*n^2+36*n-69)*x-(216*n^3+324*n^2+117*n-218)*x^2-(324*n^4-1566*n^3+297*n^2+276*n-71)*x^3)*d/dx P(n-2,x) - ((81*n^2+27*n-45)*x^2-(324*n^3-27*n-30)*x^3+(324*n^3-81*n^2-54*n+15)*x^4)*d^2/dx^2 P(n-2,x) + (9*n^2-15*n+4+(108*n^3-108*n^2+36*n-4)*x)*P(n-1,x) + ((54*n^2-9*n-12)*x-(108*n^3-36*n+8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((18*n^2 + 6*n - 4 - (216*n^3 - 216*n^2 - 252*n + 208)*x + (729*n^4 - 1134*n^3 + 81*n^2 + 216*n - 48)*x^2)*P[n - 2, x] - ((108*n^2 + 36*n - 69)*x - (216*n^3 + 324*n^2 + 117*n - 218)*x^2 - (324*n^4 - 1566*n^3 + 297*n^2 + 276*n - 71)*x^3)*D[P[n - 2, x], x] - ((81*n^2 + 27*n - 45)*x^2 - (324*n^3 - 27*n - 30)*x^3 + (324*n^3 - 81*n^2 - 54*n + 15)*x^4)*D[P[n - 2, x], {x, 2}] + (9*n^2 - 15*n + 4 + (108*n^3 - 108*n^2 + 36*n - 4)*x)*P[n - 1, x] + ((54*n^2 - 9*n - 12)*x - (108*n^3 - 36*n + 8)*x^2)*D[P[n - 1, x], x])/(9*n^2 - 6*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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((18*n^2 + 6*n - 4 - (216*n^3 - 216*n^2 - 252*n + 208)*x + (729*n^4 - 1134*n^3 + 81*n^2 + 216*n - 48)*x^2)*v[i - 2] - ((108*n^2 + 36*n - 69)*x - (216*n^3 + 324*n^2 + 117*n - 218)*x^2 - (324*n^4 - 1566*n^3 + 297*n^2 + 276*n - 71)*x^3)*deriv(v[i - 2], x) - ((81*n^2 + 27*n - 45)*x^2 - (324*n^3 - 27*n - 30)*x^3 + (324*n^3 - 81*n^2 - 54*n + 15)*x^4)*deriv(deriv(v[i - 2], x), x) + (9*n^2 - 15*n + 4 + (108*n^3 - 108*n^2 + 36*n - 4)*x)*v[i - 1] + ((54*n^2 - 9*n - 12)*x - (108*n^3 - 36*n + 8)*x^2)*deriv(v[i - 1], x))/(9*n^2 - 6*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-c77d5d4d44ca1f43","steps":["(\u03b8+1)\u00b2"],"edges":["e-d76748b3a7ac43e2"],"text":"A257610 \u2192 (\u03b8+1)\u00b2"},"headline":"A257610 \u2192 (\u03b8+1)\u00b2","chain":null}