{"id":"n-0038b0ec9e792748","revision":1,"first_index":0,"fingerprint":"f6-6897785a472f0d9f","name":"basis-map coefficients of A Jacobsthal triangle.","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"],["-3","1"],["3","-4","1"],["9","3","-5","1"],["-45","30","4","-6","1"],["81","-135","54","6","-7","1"],["27","216","-261","80","9","-8","1"],["-567","351","405","-429","107","13","-9","1"],["1539","-2862","1134","690","-644","134","18","-10","1"],["-1215","6885","-7992","2448","1122","-910","160","24","-11","1"],["-5589","-1620","18333","-17208","4326","1760","-1230","184","31","-12","1"],["24057","-44469","5103","39285","-32022","6750","2670","-1606","205","39","-13","1"],["-38637","156006","-173016","27162","74979","-54180","9640","3924","-2039","222","48","-14","1"],["-28431","-191727","555498","-480006","72927","132975","-85620","12844","5599","-2529","234","58","-15","1"],["317115","-466560","-508113","1486512","-1094337","149040","223635","-128416","16129","7776","-3075","240","69","-16","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-2","0","8","-16","0","64","-128","0","512","-1024","0","4096","-8192","0"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_coefficients_of:oeis_catalog:A113953","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["-3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160253,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-5, 3]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-5"},"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]","region_exact":null,"edges":["e-5f9e6216c74dfca3"]},{"id":160254,"position":1,"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-5f9e6216c74dfca3"]}],"region_exact":{},"incoming_edges":[{"id":"e-5f9e6216c74dfca3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-3\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-0038b0ec9e792748","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":"0126be41319432fe8cb5b172fde35f94cd9a0b4bb06e994eb1180719c04a5a9f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-5"},"upper":{"closed":true,"value":"3"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ba196dcd01800741","role":"sequence","name":"A Jacobsthal triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-8"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]"}]}],"used_by":[{"id":"e-76bf35985f9d5eb4","operation":"basis_map","operation_detail":"A027465","conclusion_id":"n-ba196dcd01800741","conclusion_name":"A Jacobsthal triangle."}],"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:    -3,     1;\n  2:     3,    -4,     1;\n  3:     9,     3,    -5,    1;\n  4:   -45,    30,     4,   -6,    1;\n  5:    81,  -135,    54,    6,   -7,    1;\n  6:    27,   216,  -261,   80,    9,   -8,   1;\n  7:  -567,   351,   405, -429,  107,   13,  -9,   1;\n  8:  1539, -2862,  1134,  690, -644,  134,  18, -10,   1;\n  9: -1215,  6885, -7992, 2448, 1122, -910, 160,  24, -11, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3-x)*P(n-1,x) - (6-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -3 + x;\nP[n_, x_] := P[n, x] = Expand[-(3 - x)*P[n - 1, x] - (6 - 2*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] = -3 + x; for(n = 2, N, my(i = n + 1); v[i] = -(3 - x)*v[i - 1] - (6 - 2*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-ba196dcd01800741","steps":["x \u21a6 x\u22123"],"edges":["e-5f9e6216c74dfca3"],"text":"A113953 \u2192 x \u21a6 x\u22123"},"headline":"A113953 \u2192 x \u21a6 x\u22123","chain":null}
