{"id":"n-fc85eff121403457","revision":0,"first_index":0,"fingerprint":"f6-5ef78deb29f8c984","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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"],["3","4"],["15","41","28"],["105","437","606","280"],["945","5298","11135","10398","3640"],["10395","73398","207253","292538","206408","58240"],["135135","1151679","4088851","7740801","8241526","4678872","1106560"],["2027025","20248155","86669703","206066737","293920512","251496652","119533584","24344320"],["34459425","394894260","1979588826","5669877104","10148317129","11623455884","8319487228","3402193040","608608000"],["654729075","8467854840","48669226974","163156152104","351574584363","504998541408","483527022084","297587310144","106824957120","17041024000"],["13749310575","198121552965","1284557266494","4935025230378","12440906986847","21503781484821","25809022247412","21238671197628","11468530713984","3669435174720","528271744000"],["316234143225","5024272277745","36280949930172","157180527359490","453931718530333","917577811703681","1324736082174390","1365996193957452","985891027140984","474327053294016","136911046052480","17961239296000"],["7905853580625","137307470242950","1092921822530865","5271908331403614","17163972475559455","39734918878214346","67068781087395947","83165139889388730","75189134854950324","48336144336566808","20972877123190592","5514739686341760","664565853952000"],["213458046676875","4023535839784650","35001188018051355","186058397849132178","674303590096249845","1759401708734165542","3400253844488264409","4928210020509333590","5356712236659207948","4312641291285316776","2499713455788818304","987813056458851200","238532865510374400","26582634158080000"],["6190283353629375","125861235360860475","1188046493634229245","6900744622401041427","27555315220303929159","80017703927429544053","174261634965728786167","289129005907766043697","367257685745018374862","355405223623620128268","257935265743144352184","136134257236225871872","49393414527531459200","11027809607530419200","1143053268797440000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","84","1428","31416","848232","27143424","1004306688","42180880896","1982501402112","103090072909824","5876134155859968","364320317663318016","24409461283442307072","1757481212407846109184"],"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":[["1","2","1"],["2","5","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187231,"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-bb62baa452376821"]},{"id":187232,"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-bb62baa452376821"]},{"id":187233,"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-bb62baa452376821"]}],"region_exact":{},"incoming_edges":[{"id":"e-bb62baa452376821","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-fc85eff121403457","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":"befed3498bb06d8f47811025072664693e0af865a180f12b0da9fb56e5e33012","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-6e1be0f1b7cdfa83","role":"sequence","name":"basis_map_A007318 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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:         3,          4;\n  2:        15,         41,          28;\n  3:       105,        437,         606,          280;\n  4:       945,       5298,       11135,        10398,         3640;\n  5:     10395,      73398,      207253,       292538,       206408,        58240;\n  6:    135135,    1151679,     4088851,      7740801,      8241526,      4678872,      1106560;\n  7:   2027025,   20248155,    86669703,    206066737,    293920512,    251496652,    119533584,     24344320;\n  8:  34459425,  394894260,  1979588826,   5669877104,  10148317129,  11623455884,   8319487228,   3402193040,    608608000;\n  9: 654729075, 8467854840, 48669226974, 163156152104, 351574584363, 504998541408, 483527022084, 297587310144, 106824957120, 17041024000;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (2*n+1+(5*n+2)*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[((2*n + 1 + (5*n + 2)*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] = ((2*n + 1 + (5*n + 2)*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-7fd16a32c438d522","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-08400314c95111bf","e-bb62baa452376821"],"text":"A059364 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A059364 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}