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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":"75f8cf89650745c7c6e5f63799f60a899c3627b45e9daa4f745656984401a75a","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-1abe254a355aa2af","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","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:           169,            624,             324;\n  1:          2197,          12168,           12636,            3456;\n  2:         28561,         210912,          328536,          179712,           32400;\n  3:        371293,        3427320,         7118280,         5840640,         2106000,          279936;\n  4:       4826809,       53466192,       138806460,       151856640,        82134000,        21835008,         2286144;\n  5:      62748517,      810903912,      2526277572,      3454738560,      2491398000,       993492864,       208039104,        17915904;\n  6:     815730721,    12047715264,     43788811248,     71858562048,     64776348000,     34441085952,     10818033408,      1863254016,      136048896;\n  7:   10604499373,   176197835736,    731898702288,   1401241959936,   1515766543200,   1007401764096,    421903302912,    109000359936,    15917720832,    1007769600;\n  8:  137858491849,  2545079849520,  11893353912180,  26023064970240,  32841608436000,  26192445866496,  13711857344640,   4723348930560,  1034651854080,  131010048000,    7316407296;\n  9: 1792160394037, 36394641848136, 188972178826860, 465162286343040, 670907143764000, 624253293151488, 392159120056704, 168859724267520, 49318405044480, 9367218432000, 1046246243328, 52242776064;\n","recurrence_sentence":"The row polynomials satisfy (169+(234*n+390)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (2197+(3042*n+9126)*x+(468*n^2+4212*n+7956)*x^2+(216*n^2+1296*n+1944)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 624*x + 324*x^2.","mathematica":"P[0, x_] := 169 + 624*x + 324*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2197 + (3042*n + 9126)*x + (468*n^2 + 4212*n + 7956)*x^2 + (216*n^2 + 1296*n + 1944)*x^3)*P[n - 1, x])/(169 + (234*n + 390)*x + (36*n^2 + 144*n + 144)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 169 + 624*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2197 + (3042*n + 9126)*x + (468*n^2 + 4212*n + 7956)*x^2 + (216*n^2 + 1296*n + 1944)*x^3)*v[i - 1])/(169 + (234*n + 390)*x + (36*n^2 + 144*n + 144)*x^2)); 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-4397e742863fdbe2","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-54bb49968ded72d0","e-e8d1779cf78e7a87"],"text":"A038305 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038305 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}