{"id":"n-31033f5999f9a349","revision":0,"first_index":0,"fingerprint":"f6-5894cfd190cff20e","name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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":[["1849","2752","1024"],["79507","177504","132096","32768"],["3418801","10176896","11360256","5636096","1048576"],["147008443","547008160","814151680","605880320","225443840","33554432"],["6321363049","28225621056","52512783360","52105707520","29082255360","8657043456","1073741824"],["271818611107","1415985322976","3161269558272","3920954490880","2917919621120","1302885040128","323196289024","34359738368"],["11688200277601","69585564443392","181246121340928","269761668972544","250941087416320","149397484601344","55589761712128","11819749998592","1099511627776"],["502592611936843","3366201679949088","10020321279848448","17399627648729088","19422840166023168","14454206635180032","7171079260864512","2287121624727552","425510999949312","35184372088832"],["21611482313284249","160829635819789760","538592268791854080","1068834269850501120","1391970211898327040","1243061770625482752","770891020542935040","327820766210949120","91484864989102080","15129279998197760","1125899906842624"],["929293739471222707","7607241774276055648","28306015904282997760","63194826204910878720","94057415746844098560","97994702917642223616","72926290543361654784","38764805604444733440","14424113713281761280","3578074719573770240","532550655936561152","36028797018963968"],["39959630797262576401","356848795956949519488","1460590420661002684416","3623170035748223713280","6066703315671444357120","7223609529357626769408","6271660986729102311424","4000527938378696491008","1860710669013347205120","615428851766688481280","137398069231632777216","18590859261785407488","1152921504606846976"],["1718264124282290785243","16623206411661231782816","74224549559045500053504","202535204998325705572352","376809683717815266181120","504749715863864170512384","500836927368795456012288","372715852925615223078912","208027452795692217532416","86006182034394715258880","25601840233494240821248","5196145163669021392896","644483121075227459584","36893488147419103232"],["73885357344138503765449","769782327678466271788864","3723598236212115919350784","11084199400817461341323264","22683942959812479024103424","33762147661116247849828352","37687978784501858064924672","32053563351602909184786432","20872087763834452492419072","10355144316941123717169152","3853076955140883243597824","1042693129509583626174464","193989419443643465334784","22209879864746300145664","1180591620717411303424"],["3177070365797955661914307","35464971525186481807415520","184747758642831905229327360","595775717793938547096125440","1330103928098095360958791680","2177658524141997986313928704","2700971812889299827986268160","2584318545222984553023406080","1923213801096174551087185920","1113178014071170799595683840","497046927213173938424119296","168134267133420359720632320","41707725180383345046978560","7162686256380681796976640","761481595362730290708480","37778931862957161709568"],["136614025729312093462315201","1626660027288553298900125184","9079032710447739342698373120","31530284141709978492471869440","76259291877624134028304056320","136202642237244964962180268032","185826860726783828165455249408","197556795457045930275567042560","165396386894271011393497989120","109409496240137930017404354560","56994714320443944939299012608","23135275157558641497559007232","7173728731025935348080312320","1642642714796636358773309440","261949668804779220003717120","25991905121714527256182784","1208925819614629174706176"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["5625","421875","31640625","2373046875","177978515625","13348388671875","1001129150390625","75084686279296875","5631351470947265625","422351360321044921875","31676352024078369140625","2375726401805877685546875","178179480135440826416015625","13363461010158061981201171875","1002259575761854648590087890625"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1849","2752","1024"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["43","32"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194997,"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-c9e24cae436e565a"]},{"id":194998,"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-7fb2b5ef0fb09dd8","e-c9e24cae436e565a"]},{"id":194999,"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-7fb2b5ef0fb09dd8","e-c9e24cae436e565a"]},{"id":195000,"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-c9e24cae436e565a"]},{"id":195001,"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-7fb2b5ef0fb09dd8"]}],"region_exact":{},"incoming_edges":[{"id":"e-7fb2b5ef0fb09dd8","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-31033f5999f9a349","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":"882c64cfbd03700e3801e83f8ce9d5c40132383c4ba59b830e527159b07e5cad","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-a8292da3a4e72c99","role":"sequence","name":"basis_map_A013611 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":{"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-c9e24cae436e565a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-31033f5999f9a349","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":"45855b4cae3c03d00aac0c79c243196180f08c437bae1e94d62b79083b0b3857","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-a8292da3a4e72c99","role":"sequence","name":"basis_map_A013611 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":{"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:               1849,                2752,                 1024;\n  1:              79507,              177504,               132096,                32768;\n  2:            3418801,            10176896,             11360256,              5636096,              1048576;\n  3:          147008443,           547008160,            814151680,            605880320,            225443840,             33554432;\n  4:         6321363049,         28225621056,          52512783360,          52105707520,          29082255360,           8657043456,           1073741824;\n  5:       271818611107,       1415985322976,        3161269558272,        3920954490880,        2917919621120,        1302885040128,         323196289024,          34359738368;\n  6:     11688200277601,      69585564443392,      181246121340928,      269761668972544,      250941087416320,      149397484601344,       55589761712128,       11819749998592,        1099511627776;\n  7:    502592611936843,    3366201679949088,    10020321279848448,    17399627648729088,    19422840166023168,    14454206635180032,     7171079260864512,     2287121624727552,      425510999949312,      35184372088832;\n  8:  21611482313284249,  160829635819789760,   538592268791854080,  1068834269850501120,  1391970211898327040,  1243061770625482752,   770891020542935040,   327820766210949120,    91484864989102080,   15129279998197760,   1125899906842624;\n  9: 929293739471222707, 7607241774276055648, 28306015904282997760, 63194826204910878720, 94057415746844098560, 97994702917642223616, 72926290543361654784, 38764805604444733440, 14424113713281761280, 3578074719573770240, 532550655936561152, 36028797018963968;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (43+32*x)*P(n-1,x) for n >= 1, with P(0,x) = 1849 + 2752*x + 1024*x^2.","mathematica":"P[0, x_] := 1849 + 2752*x + 1024*x^2;\nP[n_, x_] := P[n, x] = Expand[(43 + 32*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] = 1849 + 2752*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = (43 + 32*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 A013611","basis A007318"],"edges":["e-376f2b9433b6ff6b","e-7fb2b5ef0fb09dd8"],"text":"A038226 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A038226 \u2192 basis A013611 \u2192 basis A007318","chain":null}