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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":"561a0eaa64630aefbedb8fc6239060bb6557865e4cae13c0644cc1382306ef3b","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-89f02ea58ff2bc55","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^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:             1;\n  1:            23,             48;\n  2:           529,           2208,            1296;\n  3:         12167,          76176,           89424,           27648;\n  4:        279841,        2336064,         4113504,         2543616,          518400;\n  5:       6436343,       67161840,       157684320,       146257920,        59616000,         8957952;\n  6:     148035889,     1853666784,      5440109040,      6727864320,      4113504000,      1236197376,       146313216;\n  7:    3404825447,    49740058704,    175171511088,    270796538880,    220758048000,     99513888768,     23556427776,     2293235712;\n  8:   78310985281,  1307452971648,   5371926340032,   9965312630784,  10154870208000,   6103518511104,   2167191355392,   421955371008,   34828517376;\n  9: 1801152661463, 33830345641392, 158855536055232, 343803285762048, 420411626611200, 315857082949632, 149536203522048, 43672380899328, 7209503096832, 515978035200;\n","recurrence_sentence":"The row polynomials satisfy (529+(828*n-276)*x+144*n^2*x^2)*P(n,x) = (12167+(19044*n+19044)*x+(3312*n^2+16560*n+9936)*x^2+(1728*n^2+3456*n+1728)*x^3)*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[((12167 + (19044*n + 19044)*x + (3312*n^2 + 16560*n + 9936)*x^2 + (1728*n^2 + 3456*n + 1728)*x^3)*P[n - 1, x])/(529 + (828*n - 276)*x + 144*n^2*x^2)]];\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] = ((12167 + (19044*n + 19044)*x + (3312*n^2 + 16560*n + 9936)*x^2 + (1728*n^2 + 3456*n + 1728)*x^3)*v[i - 1])/(529 + (828*n - 276)*x + 144*n^2*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-9e472291b944d56e","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-37d2e50d1bfe4331","e-ef610e945f5f25f3"],"text":"A038326 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A038326 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}