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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"e6075f3672e47af89a926906d607cdb1481569dbc9cf58d540bc887c70417fd8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-194cb2d73cc003c5","role":"sequence","name":"basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:         1;\n   2:        13,         18;\n   3:       121,        300,         180;\n   4:      1093,       3822,        4284,        1512;\n   5:      9841,      44280,       71928,       49248,       11664;\n   6:     88573,     487146,     1035144,     1050192,      498960,       85536;\n   7:    797161,    5181540,    13601484,    18296928,    13124592,     4650048,      606528;\n   8:   7174453,   53808390,   168151140,   281932920,   271246320,   147456288,    40824000,     4199040;\n   9:  64570081,  548845680,  1989565488,  3996281664,  4829194080,  3556493568,  1532369664,   342641664,   28553472;\n  10: 581130733, 5520741954, 22773060432, 53309663136, 77603077728, 72273705408, 42560909568, 14998970880, 2776405248, 191476224;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9+36*x+36*x^2)*P(n-2,x) + (10+12*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 13 + 18*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 13 + 18*x;\nP[n_, x_] := P[n, x] = Expand[-(9 + 36*x + 36*x^2)*P[n - 2, x] + (10 + 12*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 13 + 18*x; for(n = 3, N, my(i = n); v[i] = -(9 + 36*x + 36*x^2)*v[i - 2] + (10 + 12*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":["basis A013610","basis A013609"],"edges":["e-4aa0aeb00b1923c9","e-169e30e3ff1aab79"],"text":"A091042 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A091042 \u2192 basis A013610 \u2192 basis A013609","chain":null}
