{"id":"n-538a574cb67dc127","revision":0,"first_index":0,"fingerprint":"f6-141b63b59857ebd8","name":"theta_plus_one_squared of basis_map_A007318 of Triangle whose (i,j)-th entry is 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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. 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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":"9c437ae6c994bfc8e4b470090fb8436ffef854fbe5ef4a3c9a85d76ba841b3a8","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-77900e9176a9461a","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^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:           18,            44;\n  2:          324,          1584,           1089;\n  3:         5832,         42768,          58806,          21296;\n  4:       104976,       1026432,        2117016,        1533312,         366025;\n  5:      1889568,      23094720,       63510480,       68999040,       32942250,        5797836;\n  6:     34012224,     498845952,     1714782960,     2483965440,     1778881500,      626166288,       86806489;\n  7:    612220032,   10475764992,    43212530592,    78244911360,    74713023000,    39448476144,    10937617614,     1247178944;\n  8:  11019960576,  215501451264,  1037100734208,  2253453447168,  2689668828000,  1893526854912,   787508468208,   179593767936,   17363069361;\n  9: 198359290368, 4363904388096, 24001474134528, 60843243073536, 87145270027200, 76687837623936, 42525457283232, 14547095202816, 2812817236482, 235794769100;\n","recurrence_sentence":"The row polynomials satisfy (324+(594*n-198)*x+121*n^2*x^2)*P(n,x) = (5832+(10692*n+10692)*x+(2178*n^2+10890*n+6534)*x^2+(1331*n^2+2662*n+1331)*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[((5832 + (10692*n + 10692)*x + (2178*n^2 + 10890*n + 6534)*x^2 + (1331*n^2 + 2662*n + 1331)*x^3)*P[n - 1, x])/(324 + (594*n - 198)*x + 121*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] = ((5832 + (10692*n + 10692)*x + (2178*n^2 + 10890*n + 6534)*x^2 + (1331*n^2 + 2662*n + 1331)*x^3)*v[i - 1])/(324 + (594*n - 198)*x + 121*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-b7cf68162d40a956","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-9e4137eebf66e3b4","e-0fb4d7d2ddf7c4b8"],"text":"A038277 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A038277 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}