{"id":"n-7914dcf2bf97e042","revision":1,"first_index":0,"fingerprint":"f6-3261a32f89c887ba","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"],["-2","1"],["0","-2","1"],["8","-4","-2","1"],["-16","24","-8","-2","1"],["0","-32","40","-12","-2","1"],["64","-64","-32","56","-16","-2","1"],["-128","320","-224","-16","72","-20","-2","1"],["0","-384","768","-480","16","88","-24","-2","1"],["512","-768","-384","1344","-832","64","104","-28","-2","1"],["-1024","3584","-3840","384","1984","-1280","128","120","-32","-2","1"],["0","-4096","11264","-10752","2432","2624","-1824","208","136","-36","-2","1"],["4096","-8192","-4096","23552","-22784","6272","3200","-2464","304","152","-40","-2","1"],["-8192","36864","-53248","14336","37888","-40960","12416","3648","-3200","416","168","-44","-2","1"],["0","-40960","143360","-184320","76800","49152","-66048","21376","3904","-4032","544","184","-48","-2","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-1","-1","3","-1","-5","7","3","-17","11","23","-45","-1","91","-89"],"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"],["-2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160260,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-6, 2]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-6"},"upper":{"closed":true,"value":"2"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-6, 2]","region_exact":null,"edges":["e-c345fe1d5254381d"]},{"id":160261,"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-c345fe1d5254381d"]}],"region_exact":{},"incoming_edges":[{"id":"e-c345fe1d5254381d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-2\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-7914dcf2bf97e042","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":"f2667a969fe191a44b5d18e0eb2aab20f0beecbe3477c4c6d16b264b212cdeee","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-6"},"upper":{"closed":true,"value":"2"}}},{"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-a0ae771232cbba16","operation":"basis_map","operation_detail":"A038207","conclusion_id":"n-ba196dcd01800741","conclusion_name":"A Jacobsthal triangle."}],"used_by_total":1,"links":[{"id":"l-32059740a1a159c6","kind":"oeis_match","node_a_id":"n-7914dcf2bf97e042","node_b_id":null,"fingerprint":null,"oeis_id":"A001607","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001607","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001607","relation":"match_row_sums"}],"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:   -2,    1;\n  2:    0,   -2,    1;\n  3:    8,   -4,   -2,    1;\n  4:  -16,   24,   -8,   -2,    1;\n  5:    0,  -32,   40,  -12,   -2,   1;\n  6:   64,  -64,  -32,   56,  -16,  -2,   1;\n  7: -128,  320, -224,  -16,   72, -20,  -2,   1;\n  8:    0, -384,  768, -480,   16,  88, -24,  -2,  1;\n  9:  512, -768, -384, 1344, -832,  64, 104, -28, -2, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2-x)*P(n-1,x) - (4-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -2 + x;\nP[n_, x_] := P[n, x] = Expand[-(2 - x)*P[n - 1, x] - (4 - 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] = -2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(2 - x)*v[i - 1] - (4 - 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\u22122"],"edges":["e-c345fe1d5254381d"],"text":"A113953 \u2192 x \u21a6 x\u22122"},"headline":"A113953 \u2192 x \u21a6 x\u22122","chain":null}