{"id":"n-48a9203d1b5d504a","revision":1,"first_index":1,"fingerprint":"f6-727cfd8711a24e2e","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","2"],["1","6","6"],["1","12","31","21"],["1","20","96","156","80"],["1","30","230","660","780","322"],["1","42","470","2080","4203","3898","1347"],["1","56","861","5425","16485","25578","19509","5798"],["1","72","1456","12376","52500","120288","151116","97848","25512"],["1","90","2316","25536","143892","453348","829680","874548","491868","114236"],["1","110","3510","48720","351750","1453788","3615990","5494800","4985280","2477960","518848"],["1","132","5115","87285","785466","4116420","13296822","27208170","35293500","28093472","12509112","2384538"],["1","156","7216","148500","1629540","10552080","42848652","112939200","195804895","221350140","156897642","63266436","11068567"],["1","182","9906","241956","3180320","24929190","124152600","408425160","906008103","1360526310","1362019802","869957244","320526427","51817118"],["1","210","13286","380016","5894889","55004950","329474145","1320925320","3635236605","6945740802","9189432252","8251315800","4795360297","1626415182","244370806"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","13","65","353","2023","12041","73713","461169","2935515","18950757","123780033","816513025","5431794319","36399564561"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0","0"],["-2","-6","-6","-2"],["-2","-6","-6","-2"],["4","12","12","4"]]},"terms":[{"coeff":{"coeffs":[["0","-12","-84","-216","-264","-156","-36"],["-2","56","366","862","980","546","120"],["-2","-88","-411","-806","-802","-402","-81"],["4","44","147","232","194","84","15"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","6","-72","-228","-282","-162","-36"],["0","-20","-16","238","712","848","472","102"],["0","8","-2","-118","-292","-308","-154","-30"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","2","4","-10","-40","-50","-28","-6"],["0","0","-5","-10","25","100","125","70","15"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164603,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-7103e40cdd5e6d87"]},{"id":164604,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-7103e40cdd5e6d87"]},{"id":164605,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-7103e40cdd5e6d87"]}],"region_exact":{},"incoming_edges":[{"id":"e-7103e40cdd5e6d87","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-48a9203d1b5d504a","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); 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v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^3 - 2*n^2 - 2*n + (44*n^3 - 88*n^2 + 56*n - 12)*x + (147*n^3 - 411*n^2 + 366*n - 84)*x^2 + (232*n^3 - 806*n^2 + 862*n - 216)*x^3 + (194*n^3 - 802*n^2 + 980*n - 264)*x^4 + (84*n^3 - 402*n^2 + 546*n - 156)*x^5 + (15*n^3 - 81*n^2 + 120*n - 36)*x^6)*v[i - 1] + ((8*n^2 - 20*n + 6)*x - (2*n^2 + 16*n - 6)*x^2 - (118*n^2 - 238*n + 72)*x^3 - (292*n^2 - 712*n + 228)*x^4 - (308*n^2 - 848*n + 282)*x^5 - (154*n^2 - 472*n + 162)*x^6 - (30*n^2 - 102*n + 36)*x^7)*deriv(v[i - 1], x) - ((5*n - 2)*x^2 + (10*n - 4)*x^3 - (25*n - 10)*x^4 - (100*n - 40)*x^5 - (125*n - 50)*x^6 - (70*n - 28)*x^7 - (15*n - 6)*x^8)*deriv(deriv(v[i - 1], x), x))/(4*n^3 - 2*n^2 - 2*n + (12*n^3 - 6*n^2 - 6*n)*x + (12*n^3 - 6*n^2 - 6*n)*x^2 + (4*n^3 - 2*n^2 - 2*n)*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-79ebafb266da3b41","steps":["\u2299 C(n,k+1) (shift -1)","x \u21a6 x/(x+1)"],"edges":["e-1491609a6fc5f8c9","e-7103e40cdd5e6d87"],"text":"A172431 \u2192 \u2299 C(n,k+1) (shift -1) \u2192 x \u21a6 x/(x+1)"},"headline":"A172431 \u2192 \u2299 C(n,k+1) (shift -1) \u2192 x \u21a6 x/(x+1)","chain":null}