{"id":"n-fbbb220cb64580dd","revision":0,"first_index":0,"fingerprint":"f6-d0faa4e84707c93d","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","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":true,"rows":[["121","946","1849"],["1331","15609","61017","79507"],["14641","228932","1342374","3498308","3418801"],["161051","3147815","24610190","96203470","188034055","147008443"],["1771561","41551158","406068135","2116476340","6205123815","9702557238","6321363049"],["19487171","533239861","6253449279","40742169545","159264844585","373548453663","486744954773","271818611107"],["214358881","6703586824","91717256092","717062183992","3503826580870","10957421307448","21416778010012","23920037777416","11688200277601"],["2357947691","82956886947","1297144050444","11831526035868","69375766301226","271196177359338","706753674330396","1184041869982092","1157131827482499","502592611936843"],["25937424601","1013917507130","17835730693605","185923980563640","1271889048855810","5966315901905436","19435726044085890","43414868566010040","63642250511537445","55285187313052730","21611482313284249"],["285311670611","12268401836273","239791490436245","2812100206025055","21985510701650430","120320704021759626","470344570266878538","1313299774121803710","2566904103965343615","3344753832439690165","2614989359907394129","929293739471222707"],["3138428376721","147220822035276","3165247673758434","41244136355034140","362760926577232095","2268904704410324376","10347580545871327836","34671114036815617944","84707835430856339295","147169168627346367260","172589297753888012514","122666773610201397324","39959630797262576401"],["34522712143931","1754381462587039","41148219758859642","589791149876988202","5763868055616021065","40556671591334548221","211386288294228554364","826328217877438894332","2422644093322491303837","5261297778427632629545","8226756526268661929834","8770674313129399908666","5714227204008548425343","1718264124282290785243"],["379749833583241","20782672710646462","528068820238698739","8257076098277834828","88763568056486724401","693969713896168936226","4069186049663899671507","18179220793303655675304","62181198395277276798483","162047971575571084989986","316730126261343484298609","450227948073975861978188","439995494708658228751411","264612675139472780927422","73885357344138503765449"],["4177248169415651","244938642661190445","6702411949183483995","113534796351320228885","1331453520847300866015","11450500279286787447729","74601744243838160644295","374946428861887898303145","1465699676460107238821385","4456319218328204837224615","10452094166624334981854097","18571902858051504306600255","24199752208976202581327605","21830545699006504426512315","12191083961782853121299085","3177070365797955661914307"],["45949729863572161","2873946740557967888","84258893075449513080","1537086473679412329520","19527984972427079368220","183208004468588599163664","1312990698691551627339592","7332285719965807789039280","32245392882122359254070470","112044597489394864478790320","306594762220980492801053512","653730980603412951592328976","1064789097194952913578414620","1280725347675048259688722480","1072815388636891074674319480","559164384380440196496918032","136614025729312093462315201"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["2916","157464","8503056","459165024","24794911296","1338925209984","72301961339136","3904305912313344","210832519264920576","11384956040305711104","614787626176508399616","33198531813531453579264","1792720717930698493280256","96806918768257718637133824","5227573613485916806405226496"],"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":[["121","946","1849"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["11","54","43"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195005,"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-a28a1f963e5581cf"]},{"id":195006,"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-a28a1f963e5581cf"]},{"id":195007,"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-a28a1f963e5581cf"]}],"region_exact":{},"incoming_edges":[{"id":"e-a28a1f963e5581cf","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":2,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-fbbb220cb64580dd","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":"07056913b4587fa35ed0c26de3bb48d0c5e6b11bc70e008982c0ef0465f447a6","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-a8292da3a4e72c99","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","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:          121,            946,            1849;\n  1:         1331,          15609,           61017,            79507;\n  2:        14641,         228932,         1342374,          3498308,           3418801;\n  3:       161051,        3147815,        24610190,         96203470,         188034055,          147008443;\n  4:      1771561,       41551158,       406068135,       2116476340,        6205123815,         9702557238,         6321363049;\n  5:     19487171,      533239861,      6253449279,      40742169545,      159264844585,       373548453663,       486744954773,        271818611107;\n  6:    214358881,     6703586824,     91717256092,     717062183992,     3503826580870,     10957421307448,     21416778010012,      23920037777416,      11688200277601;\n  7:   2357947691,    82956886947,   1297144050444,   11831526035868,    69375766301226,    271196177359338,    706753674330396,    1184041869982092,    1157131827482499,     502592611936843;\n  8:  25937424601,  1013917507130,  17835730693605,  185923980563640,  1271889048855810,   5966315901905436,  19435726044085890,   43414868566010040,   63642250511537445,   55285187313052730,   21611482313284249;\n  9: 285311670611, 12268401836273, 239791490436245, 2812100206025055, 21985510701650430, 120320704021759626, 470344570266878538, 1313299774121803710, 2566904103965343615, 3344753832439690165, 2614989359907394129, 929293739471222707;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (11+54*x+43*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 946*x + 1849*x^2.","mathematica":"P[0, x_] := 121 + 946*x + 1849*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((11 + 54*x + 43*x^2)*P[n - 1, x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 121 + 946*x + 1849*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((11 + 54*x + 43*x^2)*v[i - 1])/(1 + x)); 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-26633b980c3508e8","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-376f2b9433b6ff6b","e-a28a1f963e5581cf"],"text":"A038226 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A038226 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}