{"id":"n-c69ac1d7fd0ba79f","revision":0,"first_index":1,"fingerprint":"f6-110346e0d710e6d1","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Riordan array (1, 3+x).","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"],["3","9"],["10","58","84"],["33","285","819","783"],["109","1250","5368","10230","7299"],["360","5148","29412","83916","119556","68040"],["1189","20370","145263","551908","1178235","1340010","634257"],["3927","78399","670203","3180099","9045333","15422157","14593689","5912433"],["12970","295666","2946510","16766194","59578546","135383142","192109482","155637558","55114668"],["42837","1097835","12496104","82913280","353407722","1003500126","1898195040","2306451672","1633512825","513769311"],["141481","4026584","51536911","390644008","1941925538","6615165784","15638463942","25333335288","26912596869","16930294704","4789267803"],["467280","14622264","207865944","1771955424","10064121840","39988698048","113424302592","229655913552","325291712832","306971431560","173696071464","44644718160"],["1543321","52664820","823265898","7795531172","49799076639","226097990664","748096093900","1817519152392","3217939690311","4049120398212","3437079335658","1767146752980","416170266849"],["5097243","188375733","3211537914","33443272590","237332551833","1212046713423","4582741657980","12988741520340","27595679717277","43405924430979","49130778527082","37895497578558","17852498664399","3879466556121"],["16835050","669840490","12368550480","140480248220","1096438107830","6220813997502","26458530681220","85699096982824","212418017416470","400952327365062","567333035459496","583526324769660","412412969485962","179277008974290","36163709805636"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","152","1920","24256","306432","3871232","48906240","617844736","7805386752","98607398912","1245731880960","15737641762816","198817556201472","2511711808520192"],"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":[["-6","-12","-6"],["6","12","6"]]},"terms":[{"coeff":{"coeffs":[["52","306","642","574","186"],["-17","-108","-258","-260","-93"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["13","109","342","502","349","93"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184785,"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-718e7594a6b40ef0"]},{"id":184786,"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-718e7594a6b40ef0"]},{"id":184787,"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-718e7594a6b40ef0"]}],"region_exact":{},"incoming_edges":[{"id":"e-718e7594a6b40ef0","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-c69ac1d7fd0ba79f","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":"f6f121de2ba4d0c7cf287341d00551edbcfaadf3e02d5439df1bbca721213676","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6321b10e19f62245","role":"sequence","name":"basis_map_A013609 of Riordan array (1, 3+x).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     3,       9;\n   3:    10,      58,       84;\n   4:    33,     285,      819,      783;\n   5:   109,    1250,     5368,    10230,      7299;\n   6:   360,    5148,    29412,    83916,    119556,      68040;\n   7:  1189,   20370,   145263,   551908,   1178235,    1340010,     634257;\n   8:  3927,   78399,   670203,  3180099,   9045333,   15422157,   14593689,    5912433;\n   9: 12970,  295666,  2946510, 16766194,  59578546,  135383142,  192109482,  155637558,   55114668;\n  10: 42837, 1097835, 12496104, 82913280, 353407722, 1003500126, 1898195040, 2306451672, 1633512825, 513769311;\n","recurrence_sentence":"The row polynomials satisfy (6*n-6+(12*n-12)*x+(6*n-6)*x^2)*P(n,x) = -(17*n-52+(108*n-306)*x+(258*n-642)*x^2+(260*n-574)*x^3+(93*n-186)*x^4)*P(n-1,x) + (13+109*x+342*x^2+502*x^3+349*x^4+93*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(17*n - 52 + (108*n - 306)*x + (258*n - 642)*x^2 + (260*n - 574)*x^3 + (93*n - 186)*x^4)*P[n - 1, x] + (13 + 109*x + 342*x^2 + 502*x^3 + 349*x^4 + 93*x^5)*D[P[n - 1, x], x])/(6*n - 6 + (12*n - 12)*x + (6*n - 6)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(17*n - 52 + (108*n - 306)*x + (258*n - 642)*x^2 + (260*n - 574)*x^3 + (93*n - 186)*x^4)*v[i - 1] + (13 + 109*x + 342*x^2 + 502*x^3 + 349*x^4 + 93*x^5)*deriv(v[i - 1], x))/(6*n - 6 + (12*n - 12)*x + (6*n - 6)*x^2)); 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-5d39164c4108e043","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-925e2f91c3c9b55a","e-718e7594a6b40ef0"],"text":"A099097 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A099097 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}