{"id":"n-7eae85e3777acca0","revision":1,"first_index":0,"fingerprint":"f6-ca6ef6daeaeecdff","name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1872","5472","5328","1728"],["22464","88128","129600","84672","20736"],["269568","1327104","2612736","2571264","1264896","248832"],["3234816","19160064","47278080","62208000","46033920","18164736","2985984"],["38817792","268738560","797257728","1313832960","1298903040","770383872","253808640","35831808"],["465813504","3690676224","12791955456","25333088256","31352832000","24831442944","12290310144","3475685376","429981696"],["5589762048","49877876736","197791580160","457500524544","680231043072","674211299328","445461037056","189191946240","46868004864","5159780352"],["67077144576","665611665408","2972033482752","7863505256448","13652778811392","16253308108800","13436068036608","7615835799552","2832719413248","624333422592","61917364224"],["804925734912","8792265719808","43651741777920","130026464870400","258195408814080","358873043042304","356272513744896","252622846033920","125382662553600","41484634030080","8235009441792","743008370688"],["9659108818944","115166297456640","629328089972736","2084138479779840","4658662484213760","7404821422276608","8581746681446400","7306744317345792","4536066103050240","2002407559004160","596635721662464","107736213749760","8916100448256"],["115909305827328","1497904875307008","8933932649152512","32561598837030912","80913611567923200","144761806877884416","191838817244676096","190661891985506304","142113725044752384","78461683944652800","31188519367999488","8452463224946688","1399827770376192","106993205379072"],["1390911669927936","19365770173612032","125182050293514240","497946377834201088","1361702524859449344","2708105021349691392","4039207489470726144","4590008510762188800","3993307404363104256","2646904907872862208","1315802439751827456","475691791115354112","118227491943874560","18081851709063168","1283918464548864"],["16690940039135232","249080182122479616","1734573845605515264","7477541137532583936","22315786832323805184","48837690554509688832","80967750129845010432","103550592002794979328","102999790981503516672","79682547746831597568","47552488171496275968","21497930770406178816","7127031396710744064","1635712123835252736","232389242083344384","15407021574586368"],["200291280469622784","3189253465939378176","23803848332735938560","110545379797657190400","357519935638276669440","853841728642001928192","1557665288212256391168","2214220105591679877120","2478604595811581952000","2192188064740021370880","1526820431019934482432","828605027302829457408","343499546005403074560","105152922246551961600","22417216391023165440","2973555163895169024","184884258895036416"],["2403495365635473408","40674536956908011520","323917221584103800832","1612190737564717547520","5616783785231206318080","14536339971363343171584","28938084202251099832320","45262624725647235219456","56313896416839141949440","56049511926619239874560","44628101949119470239744","28265105499873167278080","14065254879698790383616","5383829619023460433920","1530841663650901524480","304689258659020013568","37901273073482465280","2218611106740436992"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["14400","345600","8294400","199065600","4777574400","114661785600","2751882854400","66045188505600","1585084524134400","38042028579225600","913008685901414400","21912208461633945600","525893003079214694400","12621432073901152665600","302914369773627663974400"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1872","5472","5328","1728"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["12","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162915,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-328ff92d3afa5881"]}],"region_exact":{},"incoming_edges":[{"id":"e-328ff92d3afa5881","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-7eae85e3777acca0","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"d8142356520cbf1026b4b02e56f3fd55c5216847e2b77016fca923563327bbf5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-c1532953a6736702","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-c1532953a6736702","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-045eb66cbb32bdb4","kind":"oeis_match","node_a_id":"n-7eae85e3777acca0","node_b_id":null,"fingerprint":null,"oeis_id":"A168702","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":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168702","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-3ff97d387cd05557","kind":"oeis_match","node_a_id":"n-7eae85e3777acca0","node_b_id":null,"fingerprint":null,"oeis_id":"A168846","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":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168846","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-4fcdca57718d2322","kind":"oeis_match","node_a_id":"n-7eae85e3777acca0","node_b_id":null,"fingerprint":null,"oeis_id":"A168894","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":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168894","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-5692589f8bc9f0ed","kind":"oeis_match","node_a_id":"n-7eae85e3777acca0","node_b_id":null,"fingerprint":null,"oeis_id":"A168798","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":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168798","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6dc749c33409e3ad","kind":"oeis_match","node_a_id":"n-7eae85e3777acca0","node_b_id":null,"fingerprint":null,"oeis_id":"A167940","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":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A167940","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bd6b66ab6d0ef9ab","kind":"oeis_match","node_a_id":"n-7eae85e3777acca0","node_b_id":null,"fingerprint":null,"oeis_id":"A168990","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":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168990","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-de3fe7cca29b2c49","kind":"oeis_match","node_a_id":"n-7eae85e3777acca0","node_b_id":null,"fingerprint":null,"oeis_id":"A168750","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":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168750","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e48a5bb6c1ea24c2","kind":"oeis_match","node_a_id":"n-7eae85e3777acca0","node_b_id":null,"fingerprint":null,"oeis_id":"A168942","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":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A168942","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A167940","relation":"match_row_sums"},{"oeis_id":"A168702","relation":"match_row_sums"},{"oeis_id":"A168750","relation":"match_row_sums"},{"oeis_id":"A168798","relation":"match_row_sums"},{"oeis_id":"A168846","relation":"match_row_sums"},{"oeis_id":"A168894","relation":"match_row_sums"},{"oeis_id":"A168942","relation":"match_row_sums"},{"oeis_id":"A168990","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:          1872,            5472,            5328,             1728;\n  1:         22464,           88128,          129600,            84672,            20736;\n  2:        269568,         1327104,         2612736,          2571264,          1264896,           248832;\n  3:       3234816,        19160064,        47278080,         62208000,         46033920,         18164736,          2985984;\n  4:      38817792,       268738560,       797257728,       1313832960,       1298903040,        770383872,        253808640,         35831808;\n  5:     465813504,      3690676224,     12791955456,      25333088256,      31352832000,      24831442944,      12290310144,       3475685376,        429981696;\n  6:    5589762048,     49877876736,    197791580160,     457500524544,     680231043072,     674211299328,     445461037056,     189191946240,      46868004864,       5159780352;\n  7:   67077144576,    665611665408,   2972033482752,    7863505256448,   13652778811392,   16253308108800,   13436068036608,    7615835799552,    2832719413248,     624333422592,     61917364224;\n  8:  804925734912,   8792265719808,  43651741777920,  130026464870400,  258195408814080,  358873043042304,  356272513744896,  252622846033920,  125382662553600,   41484634030080,   8235009441792,    743008370688;\n  9: 9659108818944, 115166297456640, 629328089972736, 2084138479779840, 4658662484213760, 7404821422276608, 8581746681446400, 7306744317345792, 4536066103050240, 2002407559004160, 596635721662464, 107736213749760, 8916100448256;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1872 + 5472*x + 5328*x^2 + 1728*x^3.","mathematica":"P[0, x_] := 1872 + 5472*x + 5328*x^2 + 1728*x^3;\nP[n_, x_] := P[n, x] = Expand[(12 + 12*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] = 1872 + 5472*x + 5328*x^2 + 1728*x^3; for(n = 1, N, my(i = n + 1); v[i] = (12 + 12*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-c1532953a6736702","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-328ff92d3afa5881"],"text":"A038338 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A038338 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}