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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":"1954d8fe946afc94ffa5179ad800d7c48e195a82f3cd1f1028ddb57558e615d6","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-c47b645304a2946e","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) = x + 4.","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:     16,      160,        144;\n  3:     64,     1184,       2664,       1024;\n  4:    256,     7712,      31968,      30848,        6400;\n  5:   1024,    46752,     313272,     556928,      292200,       36864;\n  6:   4096,   270624,    2721168,    7796992,     7558800,     2435616,      200704;\n  7:  16384,  1516960,   21807144,   93280000,   145750000,    87228576,    18582760,    1048576;\n  8:  65536,  8305696,  164974464, 1001874304,  2332000000,  2254217184,   898194304,  132891136,   5308416;\n  9: 262144, 44674208, 1195412760, 9945999488, 32744857400, 47152594656, 30459623432, 8500712960, 904652712, 26214400;\n","recurrence_sentence":"The row polynomials satisfy (n^2+6*n+9)*P(n,x) = -(12*n^2+84*n+144+(16*n^3+144*n^2+428*n+408)*x-(9*n^4+106*n^3+461*n^2+876*n+612)*x^2)*P(n-2,x) - ((8*n^2+56*n+111)*x-(16*n^3+164*n^2+557*n+642)*x^2-(4*n^4+34*n^3+77*n^2-24*n-171)*x^3)*d/dx P(n-2,x) - ((n^2+7*n+15)*x^2-(4*n^3+40*n^2+133*n+150)*x^3+(4*n^3+39*n^2+126*n+135)*x^4)*d^2/dx^2 P(n-2,x) + (7*n^2+45*n+72+(4*n^3+36*n^2+108*n+108)*x)*P(n-1,x) + ((2*n^2+13*n+24)*x-(4*n^3+40*n^2+132*n+144)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n^2 + 84*n + 144 + (16*n^3 + 144*n^2 + 428*n + 408)*x - (9*n^4 + 106*n^3 + 461*n^2 + 876*n + 612)*x^2)*P[n - 2, x] - ((8*n^2 + 56*n + 111)*x - (16*n^3 + 164*n^2 + 557*n + 642)*x^2 - (4*n^4 + 34*n^3 + 77*n^2 - 24*n - 171)*x^3)*D[P[n - 2, x], x] - ((n^2 + 7*n + 15)*x^2 - (4*n^3 + 40*n^2 + 133*n + 150)*x^3 + (4*n^3 + 39*n^2 + 126*n + 135)*x^4)*D[P[n - 2, x], {x, 2}] + (7*n^2 + 45*n + 72 + (4*n^3 + 36*n^2 + 108*n + 108)*x)*P[n - 1, x] + ((2*n^2 + 13*n + 24)*x - (4*n^3 + 40*n^2 + 132*n + 144)*x^2)*D[P[n - 1, x], x])/(n^2 + 6*n + 9)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n^2 + 84*n + 144 + (16*n^3 + 144*n^2 + 428*n + 408)*x - (9*n^4 + 106*n^3 + 461*n^2 + 876*n + 612)*x^2)*v[i - 2] - ((8*n^2 + 56*n + 111)*x - (16*n^3 + 164*n^2 + 557*n + 642)*x^2 - (4*n^4 + 34*n^3 + 77*n^2 - 24*n - 171)*x^3)*deriv(v[i - 2], x) - ((n^2 + 7*n + 15)*x^2 - (4*n^3 + 40*n^2 + 133*n + 150)*x^3 + (4*n^3 + 39*n^2 + 126*n + 135)*x^4)*deriv(deriv(v[i - 2], x), x) + (7*n^2 + 45*n + 72 + (4*n^3 + 36*n^2 + 108*n + 108)*x)*v[i - 1] + ((2*n^2 + 13*n + 24)*x - (4*n^3 + 40*n^2 + 132*n + 144)*x^2)*deriv(v[i - 1], x))/(n^2 + 6*n + 9)); 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-c47b645304a2946e","steps":["(\u03b8+1)\u00b2"],"edges":["e-b88ac48662746557"],"text":"A257606 \u2192 (\u03b8+1)\u00b2"},"headline":"A257606 \u2192 (\u03b8+1)\u00b2","chain":null}