{"id":"n-b481f693dcd8357c","revision":1,"first_index":0,"fingerprint":"f6-9160cb87e92f9c3a","name":"basis-map coefficients of Riordan array (1, 3+3x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-21","3"],["420","-123","9"],["-8379","3717","-549","27"],["167139","-99351","22122","-2187","81"],["-3333960","2484594","-739935","110079","-8181","243"],["66503241","-59590566","22229802","-4419171","493776","-29403","729"],["-1326554901","1388733255","-622605123","154960533","-23124771","2069145","-102789","2187"],["26461084860","-31692151449","16595309808","-4962494565","926874981","-110727567","8262486","-352107","6561"],["-527825129139","711775371951","-426337052967","148876328727","-33401356506","4993077492","-497328903","31827411","-1187541","19683"],["10528644929859","-15785839763379","10644826267845","-4251415746879","1113705614511","-199952793000","24917444532","-2128180635","119226492","-3956283","59049"],["-210017215815120","346557811622148","-259810466686722","116808795425826","-34994007672561","7335066799683","-1097701574733","117283764591","-8767877643","436785453","-13049829","177147"],["4189259988590481","-7544677121689932","6224794364372724","-3111201894514518","1049158482255891","-251478300314493","43933808754270","-5636619653679","527088426039","-35035366023","1571293890","-42692427","531441"],["-83564098228282581","163098233829749547","-146859019781609394","80777206773920538","-30280633263516006","8169501327561000","-1631987951314707","244414375209459","-27442739183580","2281531837158","-136518866991","5567553063","-138706101","1594323"],["1666871603033534100","-3504749205588328215","3419979403886148915","-2052888778719236430","846858985044680475","-253972907915956992","57103206075508698","-9778495294352691","1281561932575701","-127923378166458","9573388448526","-520864792659","19476249768","-448004763","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-18","306","-5184","87804","-1487160","25188408","-426622464","7225813008","-122385429792","2072873102112","-35108778101760","594646289993664","-10071675214054272","170586520633090944"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_coefficients_of:oeis_catalog:A099093","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["-21","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-21","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-21","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160157,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(17/3, 7]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"17/3"},"upper":{"closed":true,"value":"7"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (17/3, 7]","region_exact":null,"edges":["e-bc143c20b95292bc"]},{"id":160158,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":true,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign","region_exact":null,"edges":["e-bc143c20b95292bc"]},{"id":160159,"position":2,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-bc143c20b95292bc"]}],"region_exact":{},"incoming_edges":[{"id":"e-bc143c20b95292bc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-7\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-b481f693dcd8357c","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":"9fca20b3c6833998975bdad3822ae25ce6bfad0b6f20e409634367b26871252b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"17/3"},"upper":{"closed":true,"value":"7"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-016d7724d51f6416","role":"sequence","name":"Riordan array (1, 3+3x).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4/3"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]"}]}],"used_by":[{"id":"e-04516916870a85c6","operation":"basis_map","operation_detail":"A027466","conclusion_id":"n-016d7724d51f6416","conclusion_name":"Riordan array (1, 3+3x)."}],"used_by_total":1,"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:           -21,            3;\n  2:           420,         -123,             9;\n  3:         -8379,         3717,          -549,           27;\n  4:        167139,       -99351,         22122,        -2187,           81;\n  5:      -3333960,      2484594,       -739935,       110079,        -8181,        243;\n  6:      66503241,    -59590566,      22229802,     -4419171,       493776,     -29403,        729;\n  7:   -1326554901,   1388733255,    -622605123,    154960533,    -23124771,    2069145,    -102789,     2187;\n  8:   26461084860, -31692151449,   16595309808,  -4962494565,    926874981, -110727567,    8262486,  -352107,     6561;\n  9: -527825129139, 711775371951, -426337052967, 148876328727, -33401356506, 4993077492, -497328903, 31827411, -1187541, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(21-3*x)*P(n-1,x) - (21-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -21 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -21 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(21 - 3*x)*P[n - 1, x] - (21 - 3*x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -21 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(21 - 3*x)*v[i - 1] - (21 - 3*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-016d7724d51f6416","steps":["x \u21a6 x\u22127"],"edges":["e-bc143c20b95292bc"],"text":"A099093 \u2192 x \u21a6 x\u22127"},"headline":"A099093 \u2192 x \u21a6 x\u22127","chain":null}