{"id":"n-b0cf60708a253825","revision":1,"first_index":1,"fingerprint":"f6-20b7e8ee86b20d50","name":"basis-map coefficients of Riordan array (1, 3+x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-3","3"],["8","-17","9"],["-21","69","-75","27"],["55","-245","406","-297","81"],["-144","810","-1809","2007","-1107","243"],["377","-2562","7206","-10745","8964","-3969","729"],["-987","7863","-26685","50037","-56013","37449","-13851","2187"],["2584","-23611","93876","-212215","298441","-267453","149202","-47385","6561"],["-6765","69735","-317913","841551","-1425918","1604220","-1198665","573723","-159651","19683"],["17711","-203305","1045457","-3172301","6291751","-8524520","7992000","-5120577","2146176","-531441","59049"],["-46368","586548","-3358638","11493810","-26124687","41418675","-46746675","37565343","-21066885","7853517","-1751787","177147"],["121393","-1677732","10586796","-40339586","103391439","-187813815","247979530","-239823477","168629931","-84083589","28225422","-5727753","531441"],["-317811","4764459","-32848398","137926698","-393574578","806072400","-1219214685","1379097003","-1166727996","729220158","-327321729","99930591","-18600435","1594323"],["832040","-13447685","100574043","-461398906","1450772803","-3307735680","5640067910","-7307132057","7229021589","-5435130942","3057316650","-1247803785","349333884","-60052833","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","0","0","0","0","0","0","0","0","0","0","0","0","0","0"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_coefficients_of:oeis_catalog:A099097","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["-3","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160193,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(5/9, 1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"5/9"},"upper":{"closed":true,"value":"1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (5/9, 1]","region_exact":null,"edges":["e-5d14c25d8e006de4"]},{"id":160194,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":true,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n(-x) have one sign","region_exact":null,"edges":["e-5d14c25d8e006de4"]},{"id":160195,"position":2,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-5d14c25d8e006de4"]}],"region_exact":{},"incoming_edges":[{"id":"e-5d14c25d8e006de4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-b0cf60708a253825","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"35eb7e8e7dee1a9c553ad8f33a813a2d33bc6a0966d24ddd450da3614d5762be","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"5/9"},"upper":{"closed":true,"value":"1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5d39164c4108e043","role":"sequence","name":"Riordan array (1, 3+x).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-4/9"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]"}]}],"used_by":[{"id":"e-cb03f5f41d038dd3","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-5d39164c4108e043","conclusion_name":"Riordan array (1, 3+x)."}],"used_by_total":1,"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:    -3,      3;\n   3:     8,    -17,       9;\n   4:   -21,     69,     -75,      27;\n   5:    55,   -245,     406,    -297,       81;\n   6:  -144,    810,   -1809,    2007,    -1107,     243;\n   7:   377,  -2562,    7206,  -10745,     8964,   -3969,      729;\n   8:  -987,   7863,  -26685,   50037,   -56013,   37449,   -13851,   2187;\n   9:  2584, -23611,   93876, -212215,   298441, -267453,   149202, -47385,    6561;\n  10: -6765,  69735, -317913,  841551, -1425918, 1604220, -1198665, 573723, -159651, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3-3*x)*P(n-1,x) - (1-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -3 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := -3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(3 - 3*x)*P[n - 1, x] - (1 - x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -3 + 3*x; for(n = 3, N, my(i = n); v[i] = -(3 - 3*x)*v[i - 1] - (1 - x)*v[i - 2]); 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-5d39164c4108e043","steps":["x \u21a6 x\u22121"],"edges":["e-5d14c25d8e006de4"],"text":"A099097 \u2192 x \u21a6 x\u22121"},"headline":"A099097 \u2192 x \u21a6 x\u22121","chain":null}