{"id":"n-9b7571fe942d33e6","revision":0,"first_index":0,"fingerprint":"f6-4890a7f478fc1d38","name":"shift_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["729","540","100"],["19683","21870","8100","1000"],["531441","787320","437400","108000","10000"],["14348907","26572050","19683000","7290000","1350000","100000"],["387420489","860934420","797161500","393660000","109350000","16200000","1000000"],["10460353203","27119434230","30132704700","18600435000","6889050000","1530900000","189000000","10000000"],["282429536481","836828256240","1084777369200","803538792000","372008700000","110224800000","20412000000","2160000000","100000000"],["7625597484987","25418658283290","37657271530800","32543321076000","18079622820000","6696156600000","1653372000000","262440000000","24300000000","1000000000"],["205891132094649","762559748498700","1270932914164500","1255242384360000","813583026900000","361592456400000","111602610000000","23619600000000","3280500000000","270000000000","10000000000"],["5559060566555523","22648024530411390","41940786167428500","46600873519365000","34519165569900000","17898826591800000","6629195034000000","1753755300000000","324769500000000","40095000000000","2970000000000","100000000000"],["150094635296999121","667087267986662760","1358881471824683400","1677631446697140000","1398026205580950000","828459973677600000","357976531836000000","113643343440000000","26306329500000000","4330260000000000","481140000000000","32400000000000","1000000000000"],["4052555153018976267","19512302588609885730","43360672419133079400","58884863779069614000","54523022017657050000","36348681345104700000","17949966096348000000","6648135591240000000","1846704330900000000","379980315000000000","56293380000000000","5686200000000000","351000000000000","10000000000000"],["109418989131512359209","567357721422656677380","1365861181202692001100","2023498046226210372000","2060970232267436490000","1526644616494397400000","848135898052443000000","358999321926960000000","116342372846700000000","28726511814000000000","5319724410000000000","716461200000000000","66339000000000000","3780000000000000","100000000000000"],["2954312706550833698643","16412848369726853881350","42551829106699250803500","68293059060134600055000","75881176733482888950000","61829106968023094700000","38166115412359935000000","18174340672552350000000","6731237286130500000000","1939039547445000000000","430897677210000000000","72541696500000000000","8955765000000000000","765450000000000000","40500000000000000","1000000000000000"],["79766443076872509863361","472690033048133391782880","1313027869578148310508000","2269430885690626709520000","2731722362405384002200000","2428197655471452446400000","1648776185813949192000000","872368352282512800000000","363486813451047000000000","119666440642320000000000","31024632759120000000000","6267602577600000000000","967222620000000000000","110224800000000000000","8748000000000000000","432000000000000000","10000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1369","50653","1874161","69343957","2565726409","94931877133","3512479453921","129961739795077","4808584372417849","177917621779460413","6582952005840035281","243569224216081305397","9012061295995008299689","333446267951815307088493","12337511914217166362274241"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["729","540","100"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["27","10"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194152,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-b4d2f686bca92fd5"]},{"id":194153,"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 sign","region_exact":null,"edges":["e-b4d2f686bca92fd5","e-e13d2d56a91d831d"]},{"id":194154,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-b4d2f686bca92fd5","e-e13d2d56a91d831d"]},{"id":194155,"position":3,"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-b4d2f686bca92fd5"]},{"id":194156,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-e13d2d56a91d831d"]}],"region_exact":{},"incoming_edges":[{"id":"e-e13d2d56a91d831d","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-9b7571fe942d33e6","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"f8815908091c0c7f1727fd36e1a9620e7f1a6ec4baac315ce41a3cf55c85f03e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-688fdf685007c036","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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]"}]},{"id":"e-b4d2f686bca92fd5","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-9b7571fe942d33e6","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":"709524d8422bfc2deb44c3fb5f3d26172547c8d253161e0ec78d87c1c67d93ad","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-688fdf685007c036","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^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:              729,               540,               100;\n  1:            19683,             21870,              8100,              1000;\n  2:           531441,            787320,            437400,            108000,             10000;\n  3:         14348907,          26572050,          19683000,           7290000,           1350000,            100000;\n  4:        387420489,         860934420,         797161500,         393660000,         109350000,          16200000,          1000000;\n  5:      10460353203,       27119434230,       30132704700,       18600435000,        6889050000,        1530900000,        189000000,         10000000;\n  6:     282429536481,      836828256240,     1084777369200,      803538792000,      372008700000,      110224800000,      20412000000,       2160000000,       100000000;\n  7:    7625597484987,    25418658283290,    37657271530800,    32543321076000,    18079622820000,     6696156600000,    1653372000000,     262440000000,     24300000000,     1000000000;\n  8:  205891132094649,   762559748498700,  1270932914164500,  1255242384360000,   813583026900000,   361592456400000,  111602610000000,   23619600000000,   3280500000000,   270000000000,   10000000000;\n  9: 5559060566555523, 22648024530411390, 41940786167428500, 46600873519365000, 34519165569900000, 17898826591800000, 6629195034000000, 1753755300000000, 324769500000000, 40095000000000, 2970000000000, 100000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (27+10*x)*P(n-1,x) for n >= 1, with P(0,x) = 729 + 540*x + 100*x^2.","mathematica":"P[0, x_] := 729 + 540*x + 100*x^2;\nP[n_, x_] := P[n, x] = Expand[(27 + 10*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 + 540*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (27 + 10*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-a5302200d285579d","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-2f341f062a4ee4be","e-b4d2f686bca92fd5"],"text":"A038331 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A038331 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}
