{"id":"n-3c6b4dd8de6c9824","revision":0,"first_index":0,"fingerprint":"f6-036b202913a03c7f","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","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"],["27","8"],["729","432","64"],["19683","17496","5184","512"],["531441","629856","279936","55296","4096"],["14348907","21257640","12597120","3732480","552960","32768"],["387420489","688747536","510183360","201553920","44789760","5308416","262144"],["10460353203","21695547384","19284931008","9523422720","2821754880","501645312","49545216","2097152"],["282429536481","669462604992","694257516288","411411861504","152374763520","36118462464","5350883328","452984832","16777216"],["7625597484987","20334926626632","24100653779712","16662180390912","7405413507072","2194196594688","433421549568","55037657088","4076863488","134217728"],["205891132094649","610047798798960","813397065065280","642684100792320","333243607818240","118486616113152","29255954595840","4953389137920","550376570880","36238786560","1073741824"],["5559060566555523","18118419624329112","26842103147154240","23859647241914880","14139050217431040","5865087497601024","1737803702992896","367789143490560","54487280517120","5381459804160","318901321728","8589934592"],["150094635296999121","533669814389330208","869684141967797376","858947300708935680","572631533805957120","271469764174675968","93841399961616384","23832736498188288","4413469721886720","581197658849280","51662014119936","2783138807808","68719476736"],["4052555153018976267","15609842070887908584","27750830348245170816","30149050254883642368","22332629818432327680","11910735903163908096","4705475912361050112","1394215085144014848","309825574476447744","51000094564024320","6044455652032512","488440860770304","24120536334336","549755813888"],["109418989131512359209","453886177138125341904","874151155969722880704","1036030999667819710464","844173407136741986304","500250907932884140032","222333736859059617792","75287614597776801792","19519011192016207872","3855607149040238592","571201059117072384","61543548457058304","4558781367189504","207807697649664","4398046511104"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","35","1225","42875","1500625","52521875","1838265625","64339296875","2251875390625","78815638671875","2758547353515625","96549157373046875","3379220508056640625","118272717781982421875","4139545122369384765625"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["27","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191885,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-79319752a08bdbaf"]},{"id":191886,"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-79319752a08bdbaf","e-a300f2faae5b87cf"]},{"id":191887,"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-79319752a08bdbaf","e-a300f2faae5b87cf"]},{"id":191888,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-79319752a08bdbaf"]},{"id":191889,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-a300f2faae5b87cf"]}],"region_exact":{},"incoming_edges":[{"id":"e-a300f2faae5b87cf","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-3c6b4dd8de6c9824","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"31f70751ebe4206b70a5970289cd0e44be0cc4a415f33d3bf573b5e49a662230","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0d855b65b3c75e99","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","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]"}]},{"id":"e-79319752a08bdbaf","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-3c6b4dd8de6c9824","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"6979af0006d9f88dfd36bb5581618943aff539cc83928ea67d0948898381042e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0d855b65b3c75e99","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","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:            27,              8;\n  2:           729,            432,             64;\n  3:         19683,          17496,           5184,            512;\n  4:        531441,         629856,         279936,          55296,          4096;\n  5:      14348907,       21257640,       12597120,        3732480,        552960,         32768;\n  6:     387420489,      688747536,      510183360,      201553920,      44789760,       5308416,       262144;\n  7:   10460353203,    21695547384,    19284931008,     9523422720,    2821754880,     501645312,     49545216,     2097152;\n  8:  282429536481,   669462604992,   694257516288,   411411861504,  152374763520,   36118462464,   5350883328,   452984832,   16777216;\n  9: 7625597484987, 20334926626632, 24100653779712, 16662180390912, 7405413507072, 2194196594688, 433421549568, 55037657088, 4076863488, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (27+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(27 + 8*x)*P[n - 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] = (27 + 8*x)*v[i - 1]); 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-2f02b2035f8f4541","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-c22af348c5e552a2","e-79319752a08bdbaf"],"text":"A038322 \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"A038322 \u2192 basis A007318 \u2192 x \u21a6 x+1","chain":null}
