{"id":"n-8c232b5bef38e52d","revision":0,"first_index":0,"fingerprint":"f6-42aa5bf2604973b0","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.","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":false,"rows":[["1"],["6","36","46"],["36","432","1848","3312","2116"],["216","3888","28296","106272","216936","228528","97336"],["1296","31104","319680","1835136","6429024","14069376","18790080","14016384","4477456"],["7776","233280","3097440","23950080","119344320","400277376","914973120","1407732480","1395798240","805942080","205962976"],["46656","1679616","27340416","265939200","1720751040","7799763456","25389662976","59798186496","101141922240","119840083200","94456411776","44488002816","9474296896"],["279936","11757312","226654848","2657152512","21155976576","120978821376","512284010112","1631606105088","3927510744192","7110866278656","9533509888896","9180002668032","6003408824448","2387522817792","435817657216"],["1679616","80621568","1796069376","24643325952","233019472896","1609717100544","8402116552704","33796886777856","105867948713472","259109465296896","493857739597824","725386220826624","805042003885056","652727977316352","364722533308416","125515485278208","20047612231936"],["10077696","544195584","13756055040","216227045376","2367895762944","19180596621312","119081410689024","579304503189504","2238392977708032","6920559737247744","17161012829094912","34050231354138624","53661611994568704","66265646161784832","62718474561951744","43908591067324416","21416079675594240","6495426363147264","922190162669056"],["60466176","3627970560","102590945280","1817613250560","22624326743040","210274271281152","1513913147596800","8645065248522240","39763064872765440","148753512767692800","455072304490057728","1140443597885644800","2337184590854768640","3895722551065559040","5230308260946124800","5569536354076311552","4594255586327024640","2829746615597199360","1224508155126650880","331988458560860160","42420747482776576"],["362797056","23944605696","748934055936","14765840179200","205899220961280","2159731599962112","17694071681882112","116043881283551232","619440742972846080","2721664413457735680","9914661270724497408","30067926136269815808","76012403075554480128","159973386079904686080","279138352583356231680","400911552867534077952","468663022268231104512","438570353088866414592","320553684063302215680","176242568653395763200","68533484128913565696","16798616003179524096","1951354384207722496"],["2176782336","156728328192","5372298805248","116658118950912","1801416538791936","21049990202621952","193386131854147584","1432597521847320576","8708151481411497984","43967871766812229632","185962160685576388608","662531925582756642816","1994594177812367130624","5079411429467800928256","10930442555852212174848","19813225769881644367872","30085158255675000090624","37945222352096085835776","39270353839917635026944","32771624280287109414912","21501402843207630790656","10675155282716170616832","3768998572349733224448","842985093977736118272","89762301673555234816"],["13060694016","1018734133248","37976144633856","900560973791232","15255317260025856","196517210083983360","2000981599203704832","16525785427153846272","112718181740263317504","643200169936664051712","3099191315863627505664","12692351439450652409856","44372973779389955801088","132758327554858827841536","340192798975322994475008","746028212385488347201536","1396587434818990957830144","2222145416731432159199232","2985567407476014904418304","3355842712115177465905152","3115223167785792934920192","2345596150944978765250560","1395984112399431557922816","631799002173400661164032","204259971521327559991296","42008757183223849893888","4129065876983540801536"],["78364164096","6582589784064","265132088524800","6817368819695616","125699001269796864","1769720486659227648","19782253752206819328","180229841798119686144","1363282519382488793088","8677241691918540668928","46935390372953783500800","217312203824881051041792","865725295026290086821888","2977825187601921386741760","8861613379678793735995392","22829993104948063965020160","50885409007656383992086528","97927317923604731408351232","162153661448861231217868800","229834172111473480677654528","276837777386757831985348608","280590375703256581680267264","236117631826562299902885888","161943645002888963253731328","88185593077744240826302464","36668165617843565672595456","10933048343839027600588800","2081049201999704563974144","189937030341242876870656"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","88","7744","681472","59969536","5277319168","464404086784","40867559636992","3596345248055296","316478381828866048","27850097600940212224","2450808588882738675712","215671155821681003462656","18979061712307928304713728","1670157430683097690814808064"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["6","48","124","128","46"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193327,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-69534767a4cddfae"]},{"id":193328,"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-69534767a4cddfae"]},{"id":193329,"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-69534767a4cddfae"]}],"region_exact":{},"incoming_edges":[{"id":"e-69534767a4cddfae","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":2}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-8c232b5bef38e52d","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"a1e72bac85dab3ad863f588484e82ac552bb39cc3996b3ccd749e228aea49224","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7de8fc70cf185364","role":"sequence","name":"basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.","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:        6,        36,          46;\n  2:       36,       432,        1848,         3312,          2116;\n  3:      216,      3888,       28296,       106272,        216936,         228528,           97336;\n  4:     1296,     31104,      319680,      1835136,       6429024,       14069376,        18790080,        14016384,          4477456;\n  5:     7776,    233280,     3097440,     23950080,     119344320,      400277376,       914973120,      1407732480,       1395798240,        805942080,         205962976;\n  6:    46656,   1679616,    27340416,    265939200,    1720751040,     7799763456,     25389662976,     59798186496,     101141922240,     119840083200,       94456411776,       44488002816,        9474296896;\n  7:   279936,  11757312,   226654848,   2657152512,   21155976576,   120978821376,    512284010112,   1631606105088,    3927510744192,    7110866278656,     9533509888896,     9180002668032,     6003408824448,     2387522817792,      435817657216;\n  8:  1679616,  80621568,  1796069376,  24643325952,  233019472896,  1609717100544,   8402116552704,  33796886777856,  105867948713472,  259109465296896,   493857739597824,   725386220826624,   805042003885056,   652727977316352,   364722533308416,   125515485278208,    20047612231936;\n  9: 10077696, 544195584, 13756055040, 216227045376, 2367895762944, 19180596621312, 119081410689024, 579304503189504, 2238392977708032, 6920559737247744, 17161012829094912, 34050231354138624, 53661611994568704, 66265646161784832, 62718474561951744, 43908591067324416, 21416079675594240, 6495426363147264, 922190162669056;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+48*x+124*x^2+128*x^3+46*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((6 + 48*x + 124*x^2 + 128*x^3 + 46*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];\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] = ((6 + 48*x + 124*x^2 + 128*x^3 + 46*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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-d3c3090977105807","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-bced44278fba7ed1","e-69534767a4cddfae"],"text":"A272867 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A272867 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}
