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-46056243/268435456]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [\u22123\u22122\u221a2, \u22123+2\u221a2] (rational outer bound [-1564556493/268435456, -46056243/268435456])","region_exact":"[\u22123\u22122\u221a2, \u22123+2\u221a2]","edges":["e-b999f225fc465ad4"]},{"id":198443,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one 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positive length outside E works (Theorem 1); for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"faa9c770127b3cd1d996b9ac951568d3ddaf43f28f64efd9ac2de2302d04b443","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-1600c9e1b40ceecc","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 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:              729,              1120,                256;\n  1:            19683,             51632,              28928,               4096;\n  2:           531441,           2002624,            1938944,             638976,               65536;\n  3:         14348907,          70842704,          101260800,           56401920,            13041664,            1048576;\n  4:        387420489,        2369341216,         4565737216,         3728785408,          1429667840,          253755392,           16777216;\n  5:      10460353203,       76341323504,       187000432384,       206095134720,        114626723840,        33249296384,         4781506560,          268435456;\n  6:     282429536481,     2395635242368,      7161000950784,     10081431748608,       7527029145600,      3136508919808,       729003589632,        88046829568,        4294967296;\n  7:    7625597484987,    73716466969232,    261016377469952,    451420201238528,     429773489307648,    238914902163456,     79126115385344,     15312632152064,     1593432866816,      68719476736;\n  8:  205891132094649,  2234326000525792,   9162919352372480,  18908924533309440,   22125102697152512,  15625761293598720,   6863837466722304,   1880068246732800,   311320704450560,   28449863368704,   1099511627776;\n  9: 5559060566555523, 66914903242431536, 312315604647529728, 751955352333471744, 1051767584799457280, 911204725513256960, 506450740336656384, 183046230325067776, 42663039727566848, 6169291023974400, 502476813893632, 17592186044416;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(297-80*x)*P(n-2,x) + (38+16*x)*P(n-1,x) for n >= 2, with P(0,x) = 729 + 1120*x + 256*x^2, P(1,x) = 19683 + 51632*x + 28928*x^2 + 4096*x^3.","mathematica":"P[0, x_] := 729 + 1120*x + 256*x^2;\nP[1, x_] := 19683 + 51632*x + 28928*x^2 + 4096*x^3;\nP[n_, x_] := P[n, x] = Expand[-(297 - 80*x)*P[n - 2, x] + (38 + 16*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 729 + 1120*x + 256*x^2; v[2] = 19683 + 51632*x + 28928*x^2 + 4096*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(297 - 80*x)*v[i - 2] + (38 + 16*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-26633b980c3508e8","steps":["basis A013609","basis A008288"],"edges":["e-c8a2ee9cf24a53ed","e-b999f225fc465ad4"],"text":"A038226 \u2192 basis A013609 \u2192 basis A008288"},"headline":"A038226 \u2192 basis A013609 \u2192 basis A008288","chain":null}