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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":"6a5d520bb120f85ebffdcd0bf5d08d8889940fd47b793d35ee58eb22f7cd847e","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-35273d9764d27bc4","role":"sequence","name":"basis_map_A013611 of Triangular array formed by the Mersenne numbers.","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:     4,      16;\n  2:    14,     128,      144;\n  3:    48,     704,     1728,      1024;\n  4:   164,    3328,    12960,     16384,      6400;\n  5:   560,   14528,    78336,    155648,    128000,     36864;\n  6:  1912,   60416,   418176,   1146880,   1472000,    884736,    200704;\n  7:  6528,  243200,  2059776,   7249920,  12800000,  11943936,   5619712,   1048576;\n  8: 22288,  956416,  9591552,  41353216,  93568000, 119734272,  87105536,  33554432,   5308416;\n  9: 76096, 3695872, 42854400, 219119616, 607129600, 994000896, 989069312, 587202560, 191102976, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (4*n+2*n*x-(20*n^3-4*n)*x^2-20*n^3*x^3+4*n^5*x^4)*P(n,x) = (8*n+8+(14*n^2+36*n+22)*x-(40*n^3+44*n^2-24*n-28)*x^2-(8*n^4+108*n^3+168*n^2+60*n-8)*x^3+(8*n^5+8*n^4-72*n^3-144*n^2-88*n-16)*x^4+(8*n^5+24*n^4+24*n^3+8*n^2)*x^5)*P(n-1,x) + (1+6*x-(14*n^2+14*n-10)*x^2-(76*n^2+76*n)*x^3+(4*n^4+8*n^3-104*n^2-108*n-16)*x^4+(16*n^4+32*n^3-24*n^2-40*n-8)*x^5+(8*n^4+16*n^3+8*n^2)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n + 8 + (14*n^2 + 36*n + 22)*x - (40*n^3 + 44*n^2 - 24*n - 28)*x^2 - (8*n^4 + 108*n^3 + 168*n^2 + 60*n - 8)*x^3 + (8*n^5 + 8*n^4 - 72*n^3 - 144*n^2 - 88*n - 16)*x^4 + (8*n^5 + 24*n^4 + 24*n^3 + 8*n^2)*x^5)*P[n - 1, x] + (1 + 6*x - (14*n^2 + 14*n - 10)*x^2 - (76*n^2 + 76*n)*x^3 + (4*n^4 + 8*n^3 - 104*n^2 - 108*n - 16)*x^4 + (16*n^4 + 32*n^3 - 24*n^2 - 40*n - 8)*x^5 + (8*n^4 + 16*n^3 + 8*n^2)*x^6)*D[P[n - 1, x], x])/(4*n + 2*n*x - (20*n^3 - 4*n)*x^2 - 20*n^3*x^3 + 4*n^5*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n + 8 + (14*n^2 + 36*n + 22)*x - (40*n^3 + 44*n^2 - 24*n - 28)*x^2 - (8*n^4 + 108*n^3 + 168*n^2 + 60*n - 8)*x^3 + (8*n^5 + 8*n^4 - 72*n^3 - 144*n^2 - 88*n - 16)*x^4 + (8*n^5 + 24*n^4 + 24*n^3 + 8*n^2)*x^5)*v[i - 1] + (1 + 6*x - (14*n^2 + 14*n - 10)*x^2 - (76*n^2 + 76*n)*x^3 + (4*n^4 + 8*n^3 - 104*n^2 - 108*n - 16)*x^4 + (16*n^4 + 32*n^3 - 24*n^2 - 40*n - 8)*x^5 + (8*n^4 + 16*n^3 + 8*n^2)*x^6)*deriv(v[i - 1], x))/(4*n + 2*n*x - (20*n^3 - 4*n)*x^2 - 20*n^3*x^3 + 4*n^5*x^4)); 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-9b5c7e97e5191da6","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-6b7a7c8afbb896b5","e-8b8d49c7d399323e"],"text":"A110441 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A110441 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}