{"id":"n-3537990b4764b63a","revision":0,"first_index":1,"fingerprint":"f6-f25876d70728b9b9","name":"basis_map_A008277 of veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":3,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1","3","1"],["2","46","115","75","16","1"],["4","572","5258","11522","9191","3199","519","38","1"],["8","6904","203044","1158836","2254116","1933240","837459","196953","25876","1873","69","1"],["16","82928","7475336","101709864","421882396","735840732","642924568","312322780","89916277","15916483","1758593","120410","4920","109","1"],["32","995296","271102480","8466533200","70499312176","230098702016","363161608524","314806420292","162724669822","52971814038","11255806419","1595093279","152107684","9719533","407510","10686","158","1"],["64","11943872","9783576608","689261118112","11171079477936","65311538383664","175921122288192","253282904813136","214780823819264","114636891066880","40348780646538","9677219661382","1617299511711","190944775215","16017651471","952159194","39592539","1120070","20440","216","1"],["128","143327104","352495411264","55571486258496","1724037398398016","17608995740899200","78317635081883120","179708464597312208","237704006615466128","195654861538210560","105850103597525256","39174678197746248","10214618932419720","1917145125956032","262908195602455","26589829890081","1991105400072","110196362685","4469080119","130490199","2657844","35694","283","1"],["256","1719926528","12693274656896","4461230132046464","262500196381470656","4614208237618743232","33198584637879891840","118311642406816506304","236567575947086597552","289264664254323418064","230113394354733551008","124682010396056208944","47632188118438349064","13172435169027028536","2690188579591613328","411829197501075864","47758732408219721","4223462710244215","285616315250061","14752126647822","578494891530","17018515383","367984131","5653056","58203","359","1"],["512","20639120896","456999165886720","357456931847886080","39687480580754317056","1189626888286220701696","13664826296276921642688","74357820681120201551424","220139239664519822531488","390141652276439615228832","443397716917361460252560","340123915250999397378256","183094130479256365909120","71285229187120058167344","20551352197576552033256","4468866685271257044504","743487419432489554650","95666774011352559630","9594473599758445699","753664685439160971","46465884273586240","2246450228842125","84800168966520","2478239933964","55285309296","920691261","11046945","89965","444","1"],["1024","247669455872","16452465310827008","28616662717521111552","5978150013352295232256","303839818635273289231104","5523914092088590362081280","45370644229863067285629184","196055324451938392675307392","495091880812161967850929280","788042020815596819547293536","836423965872226151938482208","618017784026133077690409936","328722959032183462399209040","129246290700633848841639056","38365339339964856828296864","8744459725236750258914588","1551144078784163310565844","216422221692393283943354","23944422517494856094410","2113013093260739031991","149282711926563606543","8456950034877460471","383903543260566670","13919992717291485","400632050054148","9059962733448","158500537118","2095256416","20177245","133221","538","1"],["2048","2972033480704","592294695256720384","2290056928352059872256","898725882275367472854016","77177863480506748648540160","2208025512174582093558740736","27156581723505416747803682048","169559945247978905844177479680","602733014936968552986221050112","1324632441972367547906417479168","1913792492127423465938308881408","1905076162797699825164220909440","1355432708241324408945056462720","709654988162518889506004677456","279898441313910283593714548016","84761009133014945206565543960","20015301463249975490760928600","3732296364673415393581673100","555222187038699088517935260","66428990750614146354408940","6432289135341426957769400","506363898861914090010875","32501656810503544180585","1703061501160653191676","72813780920286389657","2533880155358694241","71430842426179621","1618892541345200","29165289766052","410811185906","4415089382","34876600","190455","641","1"],["4096","35664401788928","21322680358045485056","183230615264476420286464","134969194026123775490006016","19540528278582936469877906432","876367979741527285819866089472","16050166578701696575787469892608","143753473589831026311857437198592","712941292866349617230686686068480","2140811841886101461709185294943488","4159627653344036008064585260429824","5502437825658219546337594926929920","5156362211802442768128167575150848","3533175612316770628073353912224128","1816033548438250381744998048607104","714968971754972236013950611151584","219340572457148960484566966573728","53185214453469765333784127278216","10313624575441216031647729844760","1615078735495172975680070232100","205864215763149151239423274740","21494860834954699942992744840","1847557171956466678977221940","131199668245553326627380525","7715016071140878321490251","376042843921195414829401","15186580651079362422690","507186773954806770916","13955948958727035689","314562735690354437","5758490484904872","84583714851922","979792177806","8728903482","57581914","264394","753","1"],["8192","427972821508096","767617348835280424960","14659387419450511642152960","20258205524181959991914049536","4937982915704731527489983102976","346274486615597418763381144374272","9406577684195941098970550822552576","120217981327201145076420758615873024","826374210526218965615601880486002176","3363886537355437851179182129777098496","8710347698083243739026916636164422400","15153546935701953338319125258367501312","18486630132602647299678884754713403648","16363395718286633964529245850724836864","10803021162834047760545626952914529792","5441179074457202599771541929800711488","2130307883231558304379116381706274240","658514944443419923028012933045110336","162829721663070149703825035556706688","32561471743117436129098784380256624","5314500228915181298178999163651536","713394286467767948416906325577460","79257639977851542442571332607260","7324964433824547160880687768310","565385415061707238577117484582","36552552207138917464456958787","1983036653034903877095388871","90350312255114193294388876","3456021603623997682094941","110816376759206064007698","2969907603039731048786","66223289174981036098","1220514652496372545","18422001285422878","224826589852150","2179355351992","16355791634","91454636","358008","874","1"],["16384","5135673858179072","27634234829417811550208","1172784768723669411926024192","3039756995725681497433525170176","1246438266272025233403672529874944","136431301282077197445784364238225408","5482011907516483026335064175897464832","99586365853283083369260535389926912000","944138576613373296197514027337381416960","5178983138344152267247332328435334398464","17746820177640334822929651322101320048128","40281493988816621425001592520244971416320","63400606325405389464663354924736706472704","71769713339282896727380098303466226096896","60185560213840748324344109202804295045632","38307874554063956232036955960240881893760","18883180662391301319452793294215983367040","7331193356897850070869613639813126078912","2273816332714912350938945851738330993088","570215415951322873847984507649539878560","116803538271821641380012695562210209440","19713580547470063793806440990516334960","2761510367589318922277064294164399984","323049589149736486415681302489274616","31720968317007509108645012844476504","2625273022435397986763214875761114","183721021141676959461243055096078","10897584143534096589349290167231","548718114136474890075318332255","23469313899885041736584277215","852457246065675001595291762","26261874842490323945180119","684599934930067280017530","15045989979836722750480","277340275927635748724","4256931437427044571","53881823991977670","555010156737686","4568148156964","29274264754","140511980","474510","1004","1"]],"degrees":[2,5,8,11,14,17,20,23,26,29,32,35,38,41,44],"row_sums":["5","255","30304","6618379","2329955413","1216014303146","887654671626505","868222074880332913","1100977978580028549018","1763303973050158447840143","3491708211573773704001504083","8399360111987784791583144740504","24181742478057901824056463340302417","82268686363817064134273860583967518579","327118438360713179234442373799131477010832"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1","3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","10","7","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","10","14","3"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","7","3"]]},"deriv_order":2,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","1"]]},"deriv_order":3,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195931,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-10cf8fc9d1d5f267"]},{"id":195932,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-10cf8fc9d1d5f267"]},{"id":195933,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-10cf8fc9d1d5f267"]}],"region_exact":{},"incoming_edges":[{"id":"e-10cf8fc9d1d5f267","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-3537990b4764b63a","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"0ab81bca22fb5a18c61e3dd18df9107959e7ff7bf7c774f2c4609f765c873eb9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a8bf9bc375c66ce8","role":"sequence","name":"veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   1,           3,               1;\n   2:   2,          46,             115,                 75,                   16,                      1;\n   3:   4,         572,            5258,              11522,                 9191,                   3199,                     519,                      38,                        1;\n   4:   8,        6904,          203044,            1158836,              2254116,                1933240,                  837459,                  196953,                    25876,                     1873,                       69,                        1;\n   5:  16,       82928,         7475336,          101709864,            421882396,              735840732,               642924568,               312322780,                 89916277,                 15916483,                  1758593,                   120410,                     4920,                     109,                       1;\n   6:  32,      995296,       271102480,         8466533200,          70499312176,           230098702016,            363161608524,            314806420292,             162724669822,              52971814038,              11255806419,               1595093279,                152107684,                 9719533,                  407510,                  10686,                   158,                    1;\n   7:  64,    11943872,      9783576608,       689261118112,       11171079477936,         65311538383664,         175921122288192,         253282904813136,          214780823819264,          114636891066880,           40348780646538,            9677219661382,            1617299511711,            190944775215,             16017651471,              952159194,              39592539,              1120070,               20440,                216,                 1;\n   8: 128,   143327104,    352495411264,     55571486258496,     1724037398398016,      17608995740899200,       78317635081883120,      179708464597312208,       237704006615466128,       195654861538210560,       105850103597525256,        39174678197746248,        10214618932419720,        1917145125956032,         262908195602455,         26589829890081,         1991105400072,         110196362685,          4469080119,          130490199,           2657844,            35694,            283,             1;\n   9: 256,  1719926528,  12693274656896,   4461230132046464,   262500196381470656,    4614208237618743232,    33198584637879891840,   118311642406816506304,    236567575947086597552,    289264664254323418064,    230113394354733551008,    124682010396056208944,     47632188118438349064,    13172435169027028536,     2690188579591613328,     411829197501075864,     47758732408219721,     4223462710244215,     285616315250061,     14752126647822,      578494891530,      17018515383,      367984131,       5653056,       58203,       359,        1;\n  10: 512, 20639120896, 456999165886720, 357456931847886080, 39687480580754317056, 1189626888286220701696, 13664826296276921642688, 74357820681120201551424, 220139239664519822531488, 390141652276439615228832, 443397716917361460252560, 340123915250999397378256, 183094130479256365909120, 71285229187120058167344, 20551352197576552033256, 4468866685271257044504, 743487419432489554650, 95666774011352559630, 9594473599758445699, 753664685439160971, 46465884273586240, 2246450228842125, 84800168966520, 2478239933964, 55285309296, 920691261, 11046945, 89965, 444, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+10*x+7*x^2+x^3)*P(n-1,x) + (10*x+14*x^2+3*x^3)*d/dx P(n-1,x) + (7*x^2+3*x^3)*d^2/dx^2 P(n-1,x) + x^3*d^3/dx^3 P(n-1,x) for n >= 2, with P(1,x) = 1 + 3*x + x^2.","mathematica":"P[1, x_] := 1 + 3*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + 10*x + 7*x^2 + x^3)*P[n - 1, x] + (10*x + 14*x^2 + 3*x^3)*D[P[n - 1, x], x] + (7*x^2 + 3*x^3)*D[P[n - 1, x], {x, 2}] + x^3*D[P[n - 1, x], {x, 3}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 3*x + x^2; for(n = 2, N, my(i = n); v[i] = (2 + 10*x + 7*x^2 + x^3)*v[i - 1] + (10*x + 14*x^2 + 3*x^3)*deriv(v[i - 1], x) + (7*x^2 + 3*x^3)*deriv(deriv(v[i - 1], x), x) + x^3*deriv(deriv(deriv(v[i - 1], x), x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a8bf9bc375c66ce8","steps":["basis A008277"],"edges":["e-10cf8fc9d1d5f267"],"text":"Veronese parent of A098172 \u2192 basis A008277"},"headline":"Veronese parent of A098172 \u2192 basis A008277","chain":null}