{"id":"n-54fca3fdfcf03464","revision":1,"first_index":0,"fingerprint":"f6-8f7225f9e9fdccc1","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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":[["361","266","49"],["6859","7581","2793","343"],["130321","192052","106134","26068","2401"],["2476099","4561235","3360910","1238230","228095","16807"],["47045881","103996158","95785935","47052740","13001415","1915998","117649"],["893871739","2305248169","2547905871","1564503605","576396065","127413867","15647317","823543"],["16983563041","50056817384","64546948732","47560909592","21903050470","6455635928","1189196092","125178536","5764801"],["322687697779","1069964471583","1576789747596","1355485923372","749084326074","275978435922","67784177244","10702764828","985780971","40353607"],["6131066257801","22588138844530","37448756505405","36791760777240","23721003659010","10487180565036","3219748419090","677841772440","93649192245","7667185330","282475249"],["116490258898219","472092101850677","869643345514405","961184750305395","708241394961870","365303456348754","134585483917962","35417232609990","6524227059735","801220866985","59037327041","1977326743"],["2213314919066161","9785181747450396","19827868277728434","24350013674403340","20184879756413295","11898455435359416","5114248388882556","1615025807015544","371880942404895","60892785890860","6730255282674","450830497404","13841287201"],["42052983462257059","201411657635020651","445225769508993018","601445337757762498","553962811092675985","367364811566721969","180459907436284476","66485229055473228","18370918554801813","3760129528760605","554124351606826","55677566429394","3418797938647","96889010407"],["799006685782884121","4121192379301191782","9869171224116011899","14544041803960438588","14735410775065181201","10857671097416449306","6000291922256458827","2526438704107982664","814444055929547043","200038890930064186","36849269381853929","4936744223406268","454700125840051","25772476768262","678223072849"],["15181127029874798299","83895702007202832705","216362599913312568555","345420992844060416465","381781097353961512935","309443626276368805221","190009244204787862855","90004378833846882405","33159507991417272465","9501847319178048835","2100408354765673953","351743025917696595","43196511954804845","3672577939477335","193293575761965","4747561509943"],["288441413567621167681","1700286227345977409488","4698159312403358631480","8077537063430335892720","9671787799633691661020","8551896580728737889744","5776281023825551030792","3040147907276605805680","1260061303673856353670","412651655004303835120","106420689974794146952","21385975975795952976","3282934908565168220","372154564533703280","29380623515818680","1443258699022672","33232930569601"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976","141167095653376","3670344486987776","95428956661682176","2481152873203736576","64509974703297150976","1677259342285725925376","43608742899428874059776"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["361","266","49"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["19","7"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162060,"position":0,"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-ab827b425fb4aadf"]},{"id":162061,"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-ab827b425fb4aadf"]},{"id":162062,"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-ab827b425fb4aadf"]}],"region_exact":{},"incoming_edges":[{"id":"e-ab827b425fb4aadf","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-54fca3fdfcf03464","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":"7f5c2e0f9ab0dd184dc569f845a1998ec0963fa5800ace3721bcbfd0abb28087","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-5cf6acfdac607d6d","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^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":[{"id":"l-2de790c66753b0f2","kind":"oeis_match","node_a_id":"n-54fca3fdfcf03464","node_b_id":null,"fingerprint":null,"oeis_id":"A009970","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A009970","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009970","relation":"match_row_sums"}],"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:             361,             266,              49;\n  1:            6859,            7581,            2793,             343;\n  2:          130321,          192052,          106134,           26068,            2401;\n  3:         2476099,         4561235,         3360910,         1238230,          228095,           16807;\n  4:        47045881,       103996158,        95785935,        47052740,        13001415,         1915998,          117649;\n  5:       893871739,      2305248169,      2547905871,      1564503605,       576396065,       127413867,        15647317,         823543;\n  6:     16983563041,     50056817384,     64546948732,     47560909592,     21903050470,      6455635928,      1189196092,      125178536,       5764801;\n  7:    322687697779,   1069964471583,   1576789747596,   1355485923372,    749084326074,    275978435922,     67784177244,    10702764828,     985780971,     40353607;\n  8:   6131066257801,  22588138844530,  37448756505405,  36791760777240,  23721003659010,  10487180565036,   3219748419090,   677841772440,   93649192245,   7667185330,   282475249;\n  9: 116490258898219, 472092101850677, 869643345514405, 961184750305395, 708241394961870, 365303456348754, 134585483917962, 35417232609990, 6524227059735, 801220866985, 59037327041, 1977326743;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (19+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 266*x + 49*x^2.","mathematica":"P[0, x_] := 361 + 266*x + 49*x^2;\nP[n_, x_] := P[n, x] = Expand[(19 + 7*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] = 361 + 266*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 7*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-5cf6acfdac607d6d","steps":["basis A007318"],"edges":["e-ab827b425fb4aadf"],"text":"A038333 \u2192 basis A007318"},"headline":"A038333 \u2192 basis A007318","chain":null}