{"id":"n-a4164ed904ec2910","revision":0,"first_index":0,"fingerprint":"f6-304eeb06eba1cf88","name":"reversal of basis_map_A013611 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":[["784","1064","361"],["21952","44688","30324","6859"],["614656","1668352","1698144","768208","130321"],["17210368","58392320","79246720","53774560","18244940","2476099"],["481890304","1961981952","3328362240","3011375360","1532574960","415984632","47045881"],["13492928512","64091410432","130471799808","147557392640","100128230720","40766493936","9220992676","893871739"],["377801998336","2050925133824","4870947192832","6610571190272","5607180920320","3043898213888","1032751179712","200227269536","16983563041"],["10578455953408","64604141715456","175354098941952","277643989991424","282601918384128","191765587474944","86751099095808","25228635961536","4279857886332","322687697779"],["296196766695424","2009906631147520","6137393462968320","11105759599656960","13188089524592640","10738872898596864","6072576936706560","2354672689743360","599180104086480","90352555378120","6131066257801"],["8293509467471872","61905124239343616","210035242954915840","427571744586792960","580275939082076160","551262142127972352","374070739301124096","181309797110238720","61515824019545280","13914293528230480","1888368407402708","116490258898219"],["232218265089212416","1890920158583586816","7057184163285172224","15962678464573603840","24371589441447198720","26460582822142672896","20947961400862949376","12184018365808041984","5167329217641803520","1558400875161813760","317245892443654944","39140726989801584","2213314919066161"],["6502111422497947648","57357911477035466752","233528639585072971776","581041496110479179776","985695395187420037120","1203956518407491616768","1089293992844873367552","739163780859021213696","376181567044323296256","141814479639725052160","38492501616496799872","7123612312143888288","805646630540082604","42052983462257059"],["182059119829942534144","1729561638384454074368","7628602226445717078016","20706206043209803497472","38639259491346865455104","52438995023970745974784","53375405649398795010048","41393171728105187966976","24577195713562455355392","11118255203754444089344","3772265158416686387456","930818675453468069632","157906739585856190384","16484769517204767128","799006685782884121"],["5097655355238390956032","51886849151533622231040","246462533469784705597440","724717211512343122411520","1475317180578698499194880","2202437791006771330940928","2490852263638610433802240","2173141515725522368266240","1474631742813747321323520","778277864262811086254080","316870273307001656546304","97735960922614147311360","22106943542019866653760","3461801598613001096880","335582808028811330820","15181127029874798299"],["142734349946674946768896","1549687227992470850633728","7886801071033110579118080","24974870058271516833873920","55078508074938077303275520","89699284579184868751048704","111590181411009747434340352","108174155449448224553697280","82579377597569849994117120","49809783312819909520261120","23659647073589457022124032","8757142098666227599097856","2475977676706225065221120","516962372059541497134080","75170548998453738103680","6801144909383909637952","288441413567621167681"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2209","103823","4879681","229345007","10779215329","506623120463","23811286661761","1119130473102767","52599132235830049","2472159215084012303","116191483108948578241","5460999706120583177327","256666986187667409334369","12063348350820368238715343","566977372488557307219621121"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["784","1064","361"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["28","19"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195048,"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-c57cf1ea9141a17e"]},{"id":195049,"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-c57cf1ea9141a17e"]},{"id":195050,"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-c57cf1ea9141a17e"]}],"region_exact":{},"incoming_edges":[{"id":"e-c57cf1ea9141a17e","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-a4164ed904ec2910","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"64ab6e2cdfdee93936d72ee24716856afffaa07ee6d8826ea88295d2dc607e3e","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-3afe7d5f29fd0242","role":"sequence","name":"basis_map_A013611 of 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-5dbb25c98b33fc95","kind":"fingerprint_related","node_a_id":"n-3afe7d5f29fd0242","node_b_id":"n-a4164ed904ec2910","fingerprint":"f6-304eeb06eba1cf88","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-304eeb06eba1cf88"},"other_id":"n-3afe7d5f29fd0242","other":{"id":"n-3afe7d5f29fd0242","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","first_index":0,"fingerprint":"f6-304eeb06eba1cf88","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["361","1064","784"],["6859","30324","44688","21952"],["130321","768208","1698144","1668352","614656"]],"card_rows":[["361","1064","784"],["6859","30324","44688","21952"],["130321","768208","1698144","1668352","614656"],["2476099","18244940","53774560","79246720","58392320","17210368"]],"row_sums_preview":["2209","103823","4879681","229345007","10779215329","506623120463","23811286661761","1119130473102767"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5cf6acfdac607d6d","steps":["basis A013611"],"edges":["e-0d83dd77e9f6bf78"],"text":"A038333 \u2192 basis A013611"},"headline":"A038333 \u2192 basis A013611","chain":null}},{"id":"l-ffd623275d792afa","kind":"fingerprint_related","node_a_id":"n-ffc18199117110ab","node_b_id":"n-a4164ed904ec2910","fingerprint":"f6-304eeb06eba1cf88","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-304eeb06eba1cf88"},"other_id":"n-ffc18199117110ab","other":{"id":"n-ffc18199117110ab","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.","first_index":0,"fingerprint":"f6-304eeb06eba1cf88","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1064","1568"],["30324","89376","65856"],["768208","3396288","5005056","2458624"]],"card_rows":[["1064","1568"],["30324","89376","65856"],["768208","3396288","5005056","2458624"],["18244940","107549120","237740160","233569280","86051840"]],"row_sums_preview":["2632","185556","11628176","683155340","38529961176","2112726204484","113483578983712","6000444238763772"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5cf6acfdac607d6d","steps":["basis A013611","D"],"edges":["e-0d83dd77e9f6bf78","e-8450e20bf0b7a232"],"text":"A038333 \u2192 basis A013611 \u2192 D"},"headline":"A038333 \u2192 basis A013611 \u2192 D","chain":null}}],"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:              784,              1064,                361;\n  1:            21952,             44688,              30324,               6859;\n  2:           614656,           1668352,            1698144,             768208,             130321;\n  3:         17210368,          58392320,           79246720,           53774560,           18244940,            2476099;\n  4:        481890304,        1961981952,         3328362240,         3011375360,         1532574960,          415984632,           47045881;\n  5:      13492928512,       64091410432,       130471799808,       147557392640,       100128230720,        40766493936,         9220992676,          893871739;\n  6:     377801998336,     2050925133824,      4870947192832,      6610571190272,      5607180920320,      3043898213888,      1032751179712,       200227269536,       16983563041;\n  7:   10578455953408,    64604141715456,    175354098941952,    277643989991424,    282601918384128,    191765587474944,     86751099095808,     25228635961536,     4279857886332,      322687697779;\n  8:  296196766695424,  2009906631147520,   6137393462968320,  11105759599656960,  13188089524592640,  10738872898596864,   6072576936706560,   2354672689743360,   599180104086480,    90352555378120,    6131066257801;\n  9: 8293509467471872, 61905124239343616, 210035242954915840, 427571744586792960, 580275939082076160, 551262142127972352, 374070739301124096, 181309797110238720, 61515824019545280, 13914293528230480, 1888368407402708, 116490258898219;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (28+19*x)*P(n-1,x) for n >= 1, with P(0,x) = 784 + 1064*x + 361*x^2.","mathematica":"P[0, x_] := 784 + 1064*x + 361*x^2;\nP[n_, x_] := P[n, x] = Expand[(28 + 19*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] = 784 + 1064*x + 361*x^2; for(n = 1, N, my(i = n + 1); v[i] = (28 + 19*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 A013611","reverse"],"edges":["e-0d83dd77e9f6bf78","e-c57cf1ea9141a17e"],"text":"A038333 \u2192 basis A013611 \u2192 reverse"},"headline":"A038333 \u2192 basis A013611 \u2192 reverse","chain":null}