{"id":"n-4ec583084451043e","revision":1,"first_index":0,"fingerprint":"f6-4fed01f361177ba4","name":"basis-map coefficients of Riordan array (1, 3+3x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-9","3"],["72","-51","9"],["-567","621","-225","27"],["4455","-6615","3654","-891","81"],["-34992","65610","-48843","18063","-3321","243"],["274833","-622566","583686","-290115","80676","-11907","729"],["-2158569","5732127","-6484455","4052997","-1512351","337041","-41553","2187"],["16953624","-51637257","68435604","-51568245","24173721","-7221231","1342818","-142155","6561"],["-133155495","457531335","-695275731","613490679","-346498074","129941820","-32363955","5163507","-478953","19683"],["1045816839","-4001652315","6859243377","-6937822287","4586686479","-2071458360","647352000","-138255579","19315584","-1594323","59049"],["-8213952096","34635072852","-66108071754","75410887410","-57134690469","30194214075","-11359442025","3042792783","-568805895","70681653","-5255361","177147"],["64513217313","-297205190604","625139717004","-794004071238","678351231279","-410748813405","180777077370","-58277104911","13659024411","-2270256903","254028798","-17183259","531441"],["-506693386953","2532028855419","-5818995160506","8144433590202","-7746728418774","5288641016400","-2666422516095","1005361715187","-283514903028","59066832798","-8837686683","899375319","-55801305","1594323"],["3979621526760","-21439953492255","53449169985963","-81735432001182","85666683244347","-65106161389440","37004485557510","-15980697808659","5269956738381","-1320736818906","247642648650","-33690702195","3144004956","-180158499","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-6","30","-144","684","-3240","15336","-72576","343440","-1625184","7690464","-36391680","172207296","-814893696","3856118400"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_coefficients_of:oeis_catalog:A099093","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["-9","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-9","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-9","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160154,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(5/3, 3]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"5/3"},"upper":{"closed":true,"value":"3"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (5/3, 3]","region_exact":null,"edges":["e-7438d71d24adbb8f"]},{"id":160155,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":true,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign","region_exact":null,"edges":["e-7438d71d24adbb8f"]},{"id":160156,"position":2,"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-7438d71d24adbb8f"]}],"region_exact":{},"incoming_edges":[{"id":"e-7438d71d24adbb8f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-3\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-4ec583084451043e","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":"61273dc4961e5fbe0344fd700be9c9e3b152a7b61776c36e6409c1761bac1603","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"5/3"},"upper":{"closed":true,"value":"3"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-016d7724d51f6416","role":"sequence","name":"Riordan array (1, 3+3x).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4/3"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]"}]}],"used_by":[{"id":"e-69bbc7837aabb73c","operation":"basis_map","operation_detail":"A027465","conclusion_id":"n-016d7724d51f6416","conclusion_name":"Riordan array (1, 3+3x)."}],"used_by_total":1,"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:         -9,         3;\n  2:         72,       -51,          9;\n  3:       -567,       621,       -225,        27;\n  4:       4455,     -6615,       3654,      -891,         81;\n  5:     -34992,     65610,     -48843,     18063,      -3321,       243;\n  6:     274833,   -622566,     583686,   -290115,      80676,    -11907,       729;\n  7:   -2158569,   5732127,   -6484455,   4052997,   -1512351,    337041,    -41553,    2187;\n  8:   16953624, -51637257,   68435604, -51568245,   24173721,  -7221231,   1342818, -142155,    6561;\n  9: -133155495, 457531335, -695275731, 613490679, -346498074, 129941820, -32363955, 5163507, -478953, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9-3*x)*P(n-1,x) - (9-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -9 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -9 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(9 - 3*x)*P[n - 1, x] - (9 - 3*x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -9 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(9 - 3*x)*v[i - 1] - (9 - 3*x)*v[i - 2]); 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-016d7724d51f6416","steps":["x \u21a6 x\u22123"],"edges":["e-7438d71d24adbb8f"],"text":"A099093 \u2192 x \u21a6 x\u22123"},"headline":"A099093 \u2192 x \u21a6 x\u22123","chain":null}