{"id":"n-e7cce546ab56eb06","revision":0,"first_index":0,"fingerprint":"f6-5254523500bc8159","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)*8^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["19","27"],["361","1026","729"],["6859","29241","41553","19683"],["130321","740772","1579014","1495908","531441"],["2476099","17593335","50002110","71055630","50486895","14348907"],["47045881","401128038","1425060135","2700113940","2877753015","1635775398","387420489"],["893871739","8891671509","37906599591","89778788505","127580383665","108779063967","51526925037","10460353203"],["16983563041","193076295624","960300522972","2729275170552","4848054579270","5511472574328","3916046302812","1589973686856","282429536481"],["322687697779","4127005818963","23458769918316","77784342360732","165803466611034","235615452552522","223214639260284","135942750226188","48295450738251","7625597484987"],["6131066257801","87125678400330","557145785560005","2111289292648440","5250443109349410","8953387196995836","10602695364863490","8609707514325240","4588067820133845","1448863522147530","205891132094649"],["116490258898219","1820926678566897","12938163242449005","55157432770440495","156763229979146670","311876320695354954","443192666251293882","449857217623493790","319635391469324535","151406238064416885","43031246607781641","5559060566555523"],["2213314919066161","37742843883022956","294990121927837314","1397321630184492540","4467752054405680095","10158257302648704216","16841321317549167516","20513489123631316824","18219217313751498495","11506874092895683260","4905562113287107074","1267465809174659244","150094635296999121"],["42052983462257059","776873536592222511","6623869101470528778","34513844265556965738","122614973048689220385","313636194219278742669","594258052204949196636","844471968922822542588","900029335299324025653","710549475236308441305","403891280660638482426","156532027433070416634","37073374918358782887","4052555153018976267"],["799006685782884121","15896027748733168302","146829098415930054579","834607506785286626028","3261558283095133262241","9269691962480905061106","19759080235814560788147","32089934819067256618344","39901300531603365137283","37801232082571609077426","26858770163932459081329","13879173099065576941548","4930758864141718123971","1077979670703047687022","109418989131512359209"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","46","2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056","42420747482776576","1951354384207722496","89762301673555234816","4129065876983540801536","189937030341242876870656"],"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","1"]]},"terms":[{"coeff":{"coeffs":[["19","46","27"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191893,"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-de9d9bd90fd19532"]},{"id":191894,"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-de9d9bd90fd19532"]},{"id":191895,"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-de9d9bd90fd19532"]}],"region_exact":{},"incoming_edges":[{"id":"e-de9d9bd90fd19532","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\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-e7cce546ab56eb06","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":"74bbbeef5f614b1b72f0dcf63e90b94b3b09b22fb4669e57d061118b0d687e7a","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-0d855b65b3c75e99","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(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":[{"id":"l-66213b984362bd72","kind":"fingerprint_related","node_a_id":"n-e7cce546ab56eb06","node_b_id":"n-f891f11e3691237f","fingerprint":"f6-5254523500bc8159","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5254523500bc8159"},"other_id":"n-f891f11e3691237f","other":{"id":"n-f891f11e3691237f","name":"basis_map_A013610 of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","first_index":0,"fingerprint":"f6-5254523500bc8159","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["38","54"],["1444","4104","2916"]],"card_rows":[["1"],["38","54"],["1444","4104","2916"],["54872","233928","332424","157464"]],"row_sums_preview":["1","92","8464","778688","71639296","6590815232","606355001344","55784660123648"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4e096b5929fd3f2e","steps":["basis A013609","basis A013610"],"edges":["e-66940ce826511afb","e-f54c6d8207989193"],"text":"A038323 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038323 \u2192 basis A013609 \u2192 basis A013610","chain":null}}],"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:           19,            27;\n  2:          361,          1026,            729;\n  3:         6859,         29241,          41553,          19683;\n  4:       130321,        740772,        1579014,        1495908,          531441;\n  5:      2476099,      17593335,       50002110,       71055630,        50486895,        14348907;\n  6:     47045881,     401128038,     1425060135,     2700113940,      2877753015,      1635775398,       387420489;\n  7:    893871739,    8891671509,    37906599591,    89778788505,    127580383665,    108779063967,     51526925037,     10460353203;\n  8:  16983563041,  193076295624,   960300522972,  2729275170552,   4848054579270,   5511472574328,   3916046302812,   1589973686856,   282429536481;\n  9: 322687697779, 4127005818963, 23458769918316, 77784342360732, 165803466611034, 235615452552522, 223214639260284, 135942750226188, 48295450738251, 7625597484987;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (19+46*x+27*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[Together[((19 + 46*x + 27*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; for(n = 1, N, my(i = n + 1); v[i] = ((19 + 46*x + 27*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-2f02b2035f8f4541","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-c22af348c5e552a2","e-de9d9bd90fd19532"],"text":"A038322 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A038322 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}