{"id":"n-7b844c4754d63ea9","revision":0,"first_index":0,"fingerprint":"f6-fc9aafad96b08502","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Riordan array (1-x,x/(1-x)).","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":[["1","10","25"],["2","26","110","150"],["4","64","376","960","900"],["8","152","1136","4176","7560","5400"],["16","352","3184","15168","40176","56160","32400"],["32","800","8480","49440","171360","353376","401760","194400"],["64","1792","21760","149760","639360","1734912","2923776","2799360","1166400"],["128","3968","54272","430080","2177280","7305984","16257024","23141376","19128960","6998400"],["256","8704","132352","1185792","6935040","27675648","76349952","143824896","177106176","128770560","41990400"],["512","18944","316928","3165696","20984832","96961536","318753792","745749504","1217161728","1320178176","856604160","251942400"],["1024","40960","747520","8232960","60963840","319832064","1219276800","3404021760","6908820480","9943326720","9634277376","5643509760","1511654400"],["2048","88064","1740800","20951040","171325440","1005447168","4357545984","14123704320","34241771520","61339576320","78928515072","69092683776","36884367360","9069926400"],["4096","188416","4009984","52346880","468357120","3038846976","14747774976","54392684544","153225768960","328129781760","525894488064","611756457984","488324837376","239446056960","54419558400"],["8192","401408","9150464","128753664","1250795520","8887836672","47728631808","197272018944","632807645184","1575614177280","3020567666688","4378879844352","4647188422656","3408841138176","1545515458560","326517350400"],["16384","851968","20709376","312410112","3274113024","25280446464","148784283648","680915828736","2449247404032","6948074225664","15494820397056","26881165688832","35567655911424","34700812812288","23544077746176","9926127452160","1959104102400"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["36","288","2304","18432","147456","1179648","9437184","75497472","603979776","4831838208","38654705664","309237645312","2473901162496","19791209299968","158329674399744"],"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","10","25"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["2","8","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184398,"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-68bf18ed106e10b8"]},{"id":184399,"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-68bf18ed106e10b8"]},{"id":184400,"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-68bf18ed106e10b8"]}],"region_exact":{},"incoming_edges":[{"id":"e-68bf18ed106e10b8","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-7b844c4754d63ea9","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":"5feb2a905cc1de35083eb2e6b8313bd00a65e0f1627e1d698cb73828130510cb","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-84c761192e0db3c0","role":"sequence","name":"basis_map_A013611 of Riordan array (1-x,x/(1-x)).","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,    10,     25;\n  1:   2,    26,    110,     150;\n  2:   4,    64,    376,     960,      900;\n  3:   8,   152,   1136,    4176,     7560,     5400;\n  4:  16,   352,   3184,   15168,    40176,    56160,     32400;\n  5:  32,   800,   8480,   49440,   171360,   353376,    401760,    194400;\n  6:  64,  1792,  21760,  149760,   639360,  1734912,   2923776,   2799360,    1166400;\n  7: 128,  3968,  54272,  430080,  2177280,  7305984,  16257024,  23141376,   19128960,    6998400;\n  8: 256,  8704, 132352, 1185792,  6935040, 27675648,  76349952, 143824896,  177106176,  128770560,  41990400;\n  9: 512, 18944, 316928, 3165696, 20984832, 96961536, 318753792, 745749504, 1217161728, 1320178176, 856604160, 251942400;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (2+8*x+6*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 10*x + 25*x^2.","mathematica":"P[0, x_] := 1 + 10*x + 25*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((2 + 8*x + 6*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] = 1 + 10*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2 + 8*x + 6*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-7717563017cb9e5f","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-d3bf3843e93b142d","e-68bf18ed106e10b8"],"text":"A159854 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A159854 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}