{"id":"n-b4265c5ad843d43f","revision":0,"first_index":0,"fingerprint":"f6-b6c8acc80ab25843","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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":[["289","782","529"],["4913","19941","26979","12167"],["83521","451996","917286","827356","279841"],["1419857","9604915","25989770","35162630","23786485","6436343"],["24137569","195940266","662739135","1195529420","1213110735","656506986","148035889"],["410338673","3886148609","15773191413","35567000245","48120059155","39062165667","17616270791","3404825447"],["6975757441","75502315832","357525672028","967422406664","1636082011270","1770818176904","1197906413788","463056260792","78310985281"],["118587876497","1443981790287","7814489688612","24669271369932","50064109544862","67733795266578","61093227103188","35423803950588","11981580747993","1801152661463"],["2015993900449","27275211594310","166057905883005","599110876126920","1418483103771090","2302949039063652","2596462151885490","2007348890533320","1018434363579405","306195952448710","41426511213649"],["34271896307633","510046456813597","3450314266680215","14004216729466755","37893762915027690","71775245050817154","97107684480517326","93843560632432710","63482408663116245","28629321553954385","7746757596952363","952809757913927"],["582622237229761","9459043380906708","70386411040276386","317428912534579780","966290954333206095","2091735712909528488","3301661272337589084","3828817273803254568","3237602841818928495","1946793865668898180","790169274889141026","194373190614441108","21914624432020321"],["9904578032905937","174204048931698539","1414126985445552846","7015178967014213138","23727811211959838555","57784199069125724481","104238163026658169652","141028102918419876588","143102045608396639479","107560361078206624445","58209136583500055582","21478237562895742434","4843131999476490941","504036361936467383"],["168377826559400929","3189274126595711714","28046851878003464779","151782963104489338804","564721906844644157609","1528071042050213602942","3101085350043080547147","4794955499226275803992","5676381142466400032667","5119873187322635323582","3463443626718253307129","1703940179989728899764","576332707937702421979","119960654140879237154","11592836324538749809"],["2862423051509815793","58090350162993320505","550149786837760270665","3225387965970398449585","13091280567762205471845","38965811572280446875021","87864084917887282169165","152839206537837541252245","206782455904133144047155","217594610461212001252235","176635624962630918663579","108626186474345217359955","48988280174704705868215","15294983402962102737135","2956173262757381201295","266635235464391245607"],["48661191875666868481","1053371682955612211824","10688624429990770972920","67485040518765259868240","296735692869276657361820","963518249787298322727792","2389903109766534075001288","4619140464254645691178960","7030603500740526897603270","8455104863635666334372560","8007481664972601646082248","5909264544204379824381552","3331203051879919999038620","1386745161868563981500240","402039563735003843376120","72524784046314418805104","6132610415680998648961"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1600","64000","2560000","102400000","4096000000","163840000000","6553600000000","262144000000000","10485760000000000","419430400000000000","16777216000000000000","671088640000000000000","26843545600000000000000","1073741824000000000000000","42949672960000000000000000"],"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":[["289","782","529"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["17","40","23"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193851,"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-6700412de2b9b83a"]},{"id":193852,"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-6700412de2b9b83a"]},{"id":193853,"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-6700412de2b9b83a"]}],"region_exact":{},"incoming_edges":[{"id":"e-6700412de2b9b83a","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-b4265c5ad843d43f","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":"bb8abf31c962c3b0fa5aaf9e8f08628829a6e50ea4b3092fd2b4b916b9f79c0a","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-6ffee7a877aa82b8","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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:            289,             782,              529;\n  1:           4913,           19941,            26979,             12167;\n  2:          83521,          451996,           917286,            827356,            279841;\n  3:        1419857,         9604915,         25989770,          35162630,          23786485,           6436343;\n  4:       24137569,       195940266,        662739135,        1195529420,        1213110735,         656506986,         148035889;\n  5:      410338673,      3886148609,      15773191413,       35567000245,       48120059155,       39062165667,       17616270791,        3404825447;\n  6:     6975757441,     75502315832,     357525672028,      967422406664,     1636082011270,     1770818176904,     1197906413788,      463056260792,       78310985281;\n  7:   118587876497,   1443981790287,    7814489688612,    24669271369932,    50064109544862,    67733795266578,    61093227103188,    35423803950588,    11981580747993,     1801152661463;\n  8:  2015993900449,  27275211594310,  166057905883005,   599110876126920,  1418483103771090,  2302949039063652,  2596462151885490,  2007348890533320,  1018434363579405,   306195952448710,   41426511213649;\n  9: 34271896307633, 510046456813597, 3450314266680215, 14004216729466755, 37893762915027690, 71775245050817154, 97107684480517326, 93843560632432710, 63482408663116245, 28629321553954385, 7746757596952363, 952809757913927;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (17+40*x+23*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 782*x + 529*x^2.","mathematica":"P[0, x_] := 289 + 782*x + 529*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((17 + 40*x + 23*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] = 289 + 782*x + 529*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((17 + 40*x + 23*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-1f2dec04776afb2b","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-653ba39db5056185","e-6700412de2b9b83a"],"text":"A038320 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A038320 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}