{"id":"n-ab104f9881a1c030","revision":0,"first_index":0,"fingerprint":"f6-1798d6e40d37328a","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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"],["1","2"],["4","13","10"],["28","123","174","80"],["280","1538","3093","2714","880"],["3640","23914","61741","78584","49436","12320"],["58240","444504","1394394","2306941","2126904","1037532","209440"],["1106560","9610376","35383566","71719759","86549996","62251188","24730000","4188800"],["24344320","236879152","999477100","2391656716","3553654369","3360176044","1975837324","660943600","96342400"],["608608000","6554931120","31145785452","85777822500","151024433841","176399414694","136760510244","67895360424","19593093600","2504902400"],["17041024000","201187703360","1062174995136","3305006808108","6716241000048","9318892192821","8944877312958","5867124888948","2517572073096","638336981600","72642169600"],["528271744000","6782131572160","39365431356736","136444810895700","313963688860944","503805369978987","575495746871970","468116945572044","265792730712312","100350752768672","22678690668800","2324549427200"],["17961239296000","249081984493440","1575799271154624","6016913667939560","15450333802621596","28118111689418598","37200043342911525","36058327289968446","25421045939240148","12714671169065768","4283351829642720","872788853932800","81359229952000"],["664565853952000","9898560519505280","67769688443471808","282506178017639432","800305070078702332","1627482817008108774","2444889847885633149","2747759756759470452","2310795136770686424","1436442578946559040","641641522121279824","195060557121936960","36176267957670400","3091650738176000"],["26582634158080000","423189620792243200","3116628519038588800","14078804346887921408","43594956101871309992","97911820553551146572","164522389412757785694","210150874033689777189","205089955497965745492","152200303765460504984","84559806621660113600","34109724691849951184","9444533560306231360","1606893015791526400","126757680265216000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","27","405","8505","229635","7577955","295540245","13299311025","678264862275","38661097149675","2435649120429525","168059789309637225","12604484198222791875","1020963220056046141875"],"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":[["-2","-3","-1"],["3","6","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185833,"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-312f9799281c589c"]},{"id":185834,"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-312f9799281c589c"]},{"id":185835,"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-312f9799281c589c"]}],"region_exact":{},"incoming_edges":[{"id":"e-312f9799281c589c","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-ab104f9881a1c030","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":"ae38f6b104153bb0cf2ad4ecdd949c2f0f3c09f7d1752430d5b6061630e2a7ce","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-260e022530e5976c","role":"sequence","name":"Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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;\n  1:         1,          2;\n  2:         4,         13,          10;\n  3:        28,        123,         174,          80;\n  4:       280,       1538,        3093,        2714,          880;\n  5:      3640,      23914,       61741,       78584,        49436,        12320;\n  6:     58240,     444504,     1394394,     2306941,      2126904,      1037532,       209440;\n  7:   1106560,    9610376,    35383566,    71719759,     86549996,     62251188,     24730000,     4188800;\n  8:  24344320,  236879152,   999477100,  2391656716,   3553654369,   3360176044,   1975837324,   660943600,    96342400;\n  9: 608608000, 6554931120, 31145785452, 85777822500, 151024433841, 176399414694, 136760510244, 67895360424, 19593093600, 2504902400;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (3*n-2+(6*n-3)*x+(3*n-1)*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[((3*n - 2 + (6*n - 3)*x + (3*n - 1)*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] = ((3*n - 2 + (6*n - 3)*x + (3*n - 1)*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-260e022530e5976c","steps":["x \u21a6 x/(x+1)"],"edges":["e-312f9799281c589c"],"text":"A286718 \u2192 x \u21a6 x/(x+1)"},"headline":"A286718 \u2192 x \u21a6 x/(x+1)","chain":null}