{"id":"n-4408949914450811","revision":0,"first_index":0,"fingerprint":"f6-0723ba409d749fff","name":"basis_map_A007318 of basis_map_A013611 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","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":false,"rows":[["131","184","64"],["1561","3248","2240","512"],["18601","51280","52800","24064","4096"],["221651","760920","1041600","710656","241664","32768"],["2641211","10852968","18533760","16836608","8581120","2326528","262144"],["31472881","150604384","308187264","349598720","237424640","96534528","21757952","2097152"],["375033361","2048182688","4884548480","6643846144","5637304320","3055484928","1033109504","199229440","16777216"],["4468927451","27427854760","74691856000","118452942848","120560996352","81667719168","36819435520","10653532160","1795162112","134217728"],["53252096051","362837494040","1110840561600","2012326316032","2388718194688","1941445115904","1094141870080","422198640640","106753425408","15971909632","1073741824"],["634556225161","4752638842128","16158094835520","32916187342336","44642667868160","42325419229184","28624443932672","13808865116160","4656835067904","1045556101120","140660178944","8589934592"],["7561422605881","61745278412784","230807408201664","522246680476160","796652794961920","863104928579584","681002539155456","394279734214656","166246188318720","49785381847040","10051297214464","1228360646656","68719476736"],["90102515045411","796682082958328","3247493030886720","8080503243984896","13693164315484160","16688156083421184","15048245454569472","10165568258703360","5144535298473984","1926348532613120","518757961170944","95142115540992","10651518894080","549755813888"],["1073668757939051","10219259837450440","45112565626105600","122423736494436352","228165344942727168","308940082596347904","313403191583047680","241978503006781440","142892723462995968","64222679397302272","21625832497741824","5290540715212800","888886431580160","91809220919296","4398046511104"],["12793922580223201","130423786029959360","619857373181984000","1821904859698096128","3703680866952732672","5515915594614571008","6217310714312785920","5400802000487055360","3645396170311729152","1911887591938981888","772772667190149120","236398006448947200","52980311381770240","8212252347858944","787250325487616","35184372088832"],["152453402204739361","1657217553163847488","8436566200797377280","26699293565338589184","58791243936074833920","95511493988400365568","118421650137086951424","114306131217340170240","86808277324174196736","52041597786364313600","24546746909295902720","9013666874193346560","2526058901741240320","522297710007549952","75140624642211840","6720215068966912","281474976710656"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["379","7561","150841","3009259","60034339","1197677521","23893516081","476672644099","9509559365899","189714514673881","3784780734111721","75505900167560539","1506333222617099059","30051158552174420641","599516837820871313761"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["131","184","64"],["1561","3248","2240","512"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["12","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195762,"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-98f7ad20aa17c273"]},{"id":195763,"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-98f7ad20aa17c273"]},{"id":195764,"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-98f7ad20aa17c273"]},{"id":195765,"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-98f7ad20aa17c273"]}],"region_exact":{},"incoming_edges":[{"id":"e-98f7ad20aa17c273","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-4408949914450811","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":"3a03dcfe5a81d12a795053660512ae9c8a36096c7b05e8c158443283303606bd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-07c6a4edd51718c6","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","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:          131,           184,             64;\n  1:         1561,          3248,           2240,            512;\n  2:        18601,         51280,          52800,          24064,           4096;\n  3:       221651,        760920,        1041600,         710656,         241664,          32768;\n  4:      2641211,      10852968,       18533760,       16836608,        8581120,        2326528,         262144;\n  5:     31472881,     150604384,      308187264,      349598720,      237424640,       96534528,       21757952,        2097152;\n  6:    375033361,    2048182688,     4884548480,     6643846144,     5637304320,     3055484928,     1033109504,      199229440,      16777216;\n  7:   4468927451,   27427854760,    74691856000,   118452942848,   120560996352,    81667719168,    36819435520,    10653532160,    1795162112,     134217728;\n  8:  53252096051,  362837494040,  1110840561600,  2012326316032,  2388718194688,  1941445115904,  1094141870080,   422198640640,  106753425408,   15971909632,   1073741824;\n  9: 634556225161, 4752638842128, 16158094835520, 32916187342336, 44642667868160, 42325419229184, 28624443932672, 13808865116160, 4656835067904, 1045556101120, 140660178944, 8589934592;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (12+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 131 + 184*x + 64*x^2, P(1,x) = 1561 + 3248*x + 2240*x^2 + 512*x^3.","mathematica":"P[0, x_] := 131 + 184*x + 64*x^2;\nP[1, x_] := 1561 + 3248*x + 2240*x^2 + 512*x^3;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (12 + 8*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] = 131 + 184*x + 64*x^2; v[2] = 1561 + 3248*x + 2240*x^2 + 512*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (12 + 8*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-8eb9bb3a09f0b7fd","steps":["basis A013611","basis A007318"],"edges":["e-ab96494d3bfaa8ca","e-98f7ad20aa17c273"],"text":"A123519 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A123519 \u2192 basis A013611 \u2192 basis A007318","chain":null}