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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":"061b8dd4451650cbe6b19b0a82c7af99533483b7463d9582933731306bb9e257","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-7e380db7ce29966b","role":"sequence","name":"basis_map_A013610 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,            936,             729;\n  1:          2197,          18252,           28431,            11664;\n  2:         28561,         316368,          739206,           606528,           164025;\n  3:        371293,        5140980,        16016130,         19712160,         10661625,          2125764;\n  4:       4826809,       80199288,       312314535,        512516160,        415803375,        165809592,         26040609;\n  5:      62748517,     1216355868,      5684124537,      11659742640,      12612702375,       7544336436,       2369695419,        306110016;\n  6:     815730721,    18071572896,     98524825308,     242522646912,     327930261750,     261536996448,     123224161788,      31835441664,       3486784401;\n  7:   10604499373,   264296753604,   1646772080148,    4729191614784,    7673568124950,    7649957146104,    4805742309732,    1862373337344,     407953774917,     38742048900;\n  8:  137858491849,  3817619774280,  26760046302405,   87827844274560,  166260642707250,  198898885798704,  156186625066290,   80702844618240,   26516995369605,   5036466357000,   421900912521;\n  9: 1792160394037, 54591962772204, 425187402360435, 1569922716407760, 3396467415305250, 4740423444869112, 4466937476895894, 2885126695102080, 1263976779284505, 360107344525500, 60331830490503, 4518872583696;\n","recurrence_sentence":"The row polynomials satisfy (169+(351*n+585)*x+(81*n^2+324*n+324)*x^2)*P(n,x) = (2197+(4563*n+13689)*x+(1053*n^2+9477*n+17901)*x^2+(729*n^2+4374*n+6561)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 936*x + 729*x^2.","mathematica":"P[0, x_] := 169 + 936*x + 729*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2197 + (4563*n + 13689)*x + (1053*n^2 + 9477*n + 17901)*x^2 + (729*n^2 + 4374*n + 6561)*x^3)*P[n - 1, x])/(169 + (351*n + 585)*x + (81*n^2 + 324*n + 324)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 169 + 936*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2197 + (4563*n + 13689)*x + (1053*n^2 + 9477*n + 17901)*x^2 + (729*n^2 + 4374*n + 6561)*x^3)*v[i - 1])/(169 + (351*n + 585)*x + (81*n^2 + 324*n + 324)*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 A013610","(\u03b8+1)\u00b2"],"edges":["e-c80be26a429cc3fc","e-4699ad5ea55d9d96"],"text":"A038305 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A038305 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}