{"id":"n-28cc1c51f0f8e06b","revision":0,"first_index":0,"fingerprint":"f6-e0f3e80aae5e0ae7","name":"mobius_x_over_x_plus_one of basis_map_A013610 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","6"],["15","63","66"],["105","681","1470","1056"],["945","8334","27531","40374","22176"],["10395","116244","519525","1159920","1293660","576576"],["135135","1833417","10357389","31184235","52775100","47598948","17873856"],["2027025","32366115","221363847","840629529","1914258960","2613887820","1981669968","643458816"],["34459425","633331980","5090196114","23366619720","67008213009","122920710300","140857790076","92187268560","26381811456"],["654729075","13618441170","125846997246","678114795924","2348020554291","5417871294114","8330650685244","8231016446136","4741868771424","1213563326976"],["13749310575","319378447395","3337327441836","20658607573950","83892286232235","233524345445235","451255100389938","597714530316300","519361082952840","267320137209120","61891729675776"],["316234143225","8115665682285","94643724216348","662038310943666","3086404607482605","10069027974245565","23456230653901734","39017719819111428","45417318605628120","35232583801168800","16393437466253568","3465936861843456"],["7905853580625","222181924793850","2861148712028085","22324224950788878","117544452154628751","439996380412578030","1200616472776522815","2406273065365791474","3515513874106500108","3651271029972535320","2559023548527636000","1086648106987554048","211422148572450816"],["213458046676875","6520698305755200","91915022261152395","791589888665153316","4647099054927042225","19637836113345104376","61456405872196265985","144210059968126875588","253732896915017740200","330608233500287460768","310077523788433503120","198235053091487935296","77427173072634739200","13953861805781753856"],["6190283353629375","204255772180958925","3128505225282038655","29482073351831266209","190968754688110109961","899441280186828024879","3176522134341194124261","8545496556001614276987","17597168268272522632548","27602674452474595916472","32465432768384981305008","27764320728631928845104","16320076788602050842816","5901991280524737345024","990724188210504523776"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","144","3312","99360","3676320","161758080","8249662080","478480400640","31101226041600","2239288274995200","176903773724620800","15213724540317388800","1414876382249517158400","141487638224951715840000"],"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","7","5"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190517,"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-d51e1af424cea21e"]},{"id":190518,"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-d51e1af424cea21e"]},{"id":190519,"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-d51e1af424cea21e"]}],"region_exact":{},"incoming_edges":[{"id":"e-d51e1af424cea21e","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-28cc1c51f0f8e06b","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":"278429fedd2ed0e117cd3bf8718e42d02343bbf24a523955550e2eb6c6fed1c9","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-629fc911ba71fa1d","role":"sequence","name":"basis_map_A013610 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,           6;\n  2:        15,          63,           66;\n  3:       105,         681,         1470,         1056;\n  4:       945,        8334,        27531,        40374,         22176;\n  5:     10395,      116244,       519525,      1159920,       1293660,        576576;\n  6:    135135,     1833417,     10357389,     31184235,      52775100,      47598948,      17873856;\n  7:   2027025,    32366115,    221363847,    840629529,    1914258960,    2613887820,    1981669968,     643458816;\n  8:  34459425,   633331980,   5090196114,  23366619720,   67008213009,  122920710300,  140857790076,   92187268560,   26381811456;\n  9: 654729075, 13618441170, 125846997246, 678114795924, 2348020554291, 5417871294114, 8330650685244, 8231016446136, 4741868771424, 1213563326976;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (2*n+1+(7*n+2)*x+(5*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 + (7*n + 2)*x + (5*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 + (7*n + 2)*x + (5*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 A013610","x \u21a6 x/(x+1)"],"edges":["e-07b12f78e33af42b","e-d51e1af424cea21e"],"text":"A059364 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A059364 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}
