{"id":"n-86e0f3639d6a7bfe","revision":0,"first_index":0,"fingerprint":"f6-4305ea87645379bc","name":"shift_x_plus_one of basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["28","30","8"],["784","1680","1348","480","64"],["21952","70560","94416","67320","26976","5760","512"],["614656","2634240","4936064","5281920","3530256","1509120","402944","61440","4096"],["17210368","92198400","222154240","317049600","296793280","190418400","84798080","25881600","5181440","614400","32768"],["481890304","3097866240","9123953664","16279603200","19598933760","16771910400","10461244480","4791974400","1599912960","379699200","60801024","5898240","262144"],["13492928512","101196963840","352261812224","754330429440","1110149870592","1187818329600","952863627520","582187900800","272246750720","96964761600","25892708352","5026775040","670695424","55050240","2097152"],["377801998336","3238302842880","13007183085568","32498682101760","56532203757568","72598052782080","71195930423296","54389716684800","32711455064320","15539919052800","5811912687616","1693249044480","376724389888","61876469760","7075790848","503316480","16777216"],["10578455953408","102006539550720","464372627668992","1326085014159360","2661919632949248","3988701047439360","4623685265375232","4239574402129920","3117179685731328","1851579118886400","890622767351808","346087706296320","107841055752192","26580265205760","5068211355648","721378344960","72175583232","4529848320","134217728"],["296196766695424","3173536786022400","16147257408880640","51877611542937600","118031281168711680","202149898430054400","270419575917281280","289328249600409600","251457745387376640","179276201117798400","105422248538293248","51221771747942400","20527162888765440","6748180748697600","1802046320148480","384887055974400","64207957524480","8063130009600","717058211840","40265318400","1073741824"],["8293509467471872","97744933009489920","549698885162893312","1962379139756851200","4990382278283100160","9616156483446374400","14580495327935201280","17830977453770342400","17884020966197084160","14878091989722931200","10341770955705176064","6031086674033571840","2954791701630050304","1214538121610035200","417120022535208960","118823673161318400","27760805051105280","5231102813798400","775635190415360","87144215347200","6974490017792","354334801920","8589934592"],["232218265089212416","2985663408289873920","18390264850585485312","72219541932154552320","202999669065935486976","434662883003046297600","736661621911841669120","1013611480411196620800","1152325873290110238720","1095755024328317337600","878984514181009539072","598148291461878743040","346400935512289972224","170899511846251069440","71753837892327309312","25556968497453465600","7678972916552171520","1929884415610060800","400737310154424320","67557916252569600","9014693704237056","916309797765120","66666482368512","3092376453120","68719476736"],["6502111422497947648","90565123384792842240","606355064185803505664","2597740426884211015680","7997699110615514087424","18838307131520597360640","35290409256150439559168","53958273172893125836800","68566761839753718661120","73359808723185765580800","66702824113238669131776","51883727780989429678080","34675551051648557826048","19962400728758759055360","9907300300471016521728","4235406349468524871680","1555750999725683245056","488861699113274572800","130548960485043077120","29352816284585164800","5485047392531120128","836560815485091840","101473505073168384","9417059393863680","628027297890304","26800595927040","549755813888"],["182059119829942534144","2730886797449138012160","19746912390126267006976","91651904865410356346880","306718628417247152963584","788185496416115836846080","1617262266002754341175296","2720250383577685489090560","3820940800749101414088704","4542743684824958002790400","4617007431584285452402688","4040707571050350229585920","3061049855781751862525952","2014283574202617425756160","1153680838689139696877568","575509592629319264501760","249881620880143009185792","94243908362690384363520","30767229864784909303808","8649241263425873510400","2078557499408770924544","422797654886197493760","71818461746850430976","10000369339103969280","1111884586742185984","94927710773575680","5843629423722496","230897441832960","4398046511104"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","66","4356","287496","18974736","1252332576","82653950016","5455160701056","360040606269696","23762680013799936","1568336880910795776","103510234140112521216","6831675453247426400256","450890579914330142416896","29758778274345789399515136"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["28","30","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192053,"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-43e019f3ea1efe23"]},{"id":192054,"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-06731d865faeae24","e-43e019f3ea1efe23"]},{"id":192055,"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-06731d865faeae24","e-43e019f3ea1efe23"]},{"id":192056,"position":3,"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-06731d865faeae24"]}],"region_exact":{},"incoming_edges":[{"id":"e-06731d865faeae24","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-86e0f3639d6a7bfe","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":"418f5351d2e26c9083e80cfd8988dbe78d76fbaa576b026492d4e98ae678aee3","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-83675781d9687218","role":"sequence","name":"basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^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]"}]},{"id":"e-43e019f3ea1efe23","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-86e0f3639d6a7bfe","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":"bbf87544f00609bfd86e415963e6a39bb2927fbdd7c78d49ad8fa6f05838863c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-83675781d9687218","role":"sequence","name":"basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^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:              1;\n  1:             28,              30,               8;\n  2:            784,            1680,            1348,              480,               64;\n  3:          21952,           70560,           94416,            67320,            26976,             5760,              512;\n  4:         614656,         2634240,         4936064,          5281920,          3530256,          1509120,           402944,            61440,             4096;\n  5:       17210368,        92198400,       222154240,        317049600,        296793280,        190418400,         84798080,         25881600,          5181440,           614400,           32768;\n  6:      481890304,      3097866240,      9123953664,      16279603200,      19598933760,      16771910400,      10461244480,       4791974400,       1599912960,        379699200,        60801024,         5898240,          262144;\n  7:    13492928512,    101196963840,    352261812224,     754330429440,    1110149870592,    1187818329600,     952863627520,     582187900800,     272246750720,      96964761600,     25892708352,      5026775040,       670695424,       55050240,       2097152;\n  8:   377801998336,   3238302842880,  13007183085568,   32498682101760,   56532203757568,   72598052782080,   71195930423296,   54389716684800,   32711455064320,   15539919052800,   5811912687616,   1693249044480,    376724389888,    61876469760,    7075790848,    503316480,    16777216;\n  9: 10578455953408, 102006539550720, 464372627668992, 1326085014159360, 2661919632949248, 3988701047439360, 4623685265375232, 4239574402129920, 3117179685731328, 1851579118886400, 890622767351808, 346087706296320, 107841055752192, 26580265205760, 5068211355648, 721378344960, 72175583232, 4529848320, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (28+30*x+8*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(28 + 30*x + 8*x^2)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (28 + 30*x + 8*x^2)*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-eed4f744ce063213","steps":["basis A013609","basis A007318"],"edges":["e-185113cbc4b6c77b","e-06731d865faeae24"],"text":"A200536 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A200536 \u2192 basis A013609 \u2192 basis A007318","chain":null}