{"id":"n-54865b1c531f7155","revision":1,"first_index":0,"fingerprint":"f6-074a9ec3b69568bd","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"],["-5","1"],["15","-8","1"],["-25","35","-11","1"],["-25","-90","64","-14","1"],["375","25","-230","102","-17","1"],["-1625","1100","355","-472","149","-20","1"],["4375","-6625","1675","1235","-843","205","-23","1"],["-5625","23250","-16350","930","3016","-1370","270","-26","1"],["-15625","-46875","75000","-30000","-3250","6130","-2080","344","-29","1"],["134375","-25000","-211875","183000","-42050","-14168","11090","-3000","427","-32","1"],["-515625","696875","190625","-676875","365750","-39010","-36558","18490","-4157","519","-35","1"],["1234375","-3481250","1812500","1321250","-1719125","618380","4544","-76828","29005","-5578","620","-38","1"],["-1015625","10640625","-13056250","2356250","4905625","-3689425","883220","130668","-143303","43391","-7290","730","-41","1"],["-7265625","-16937500","50834375","-34425000","-2338125","13730700","-6914205","1007248","403477","-246468","62485","-9320","849","-44","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-4","8","0","-64","256","-512","0","4096","-16384","32768","0","-262144","1048576","-2097152"],"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"],["-5","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-10","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160264,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-3, 5]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":true,"value":"5"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-3, 5]","region_exact":null,"edges":["e-0c61b86c2e25257b"]},{"id":160265,"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-0c61b86c2e25257b"]}],"region_exact":{},"incoming_edges":[{"id":"e-0c61b86c2e25257b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-5\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-54865b1c531f7155","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":"4f5d14bbf6302830426f68b1e7dc69e949d8696981099ab3c1b4adf3566c6a91","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":true,"value":"5"}}},{"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-7b55c4767302dd2a","operation":"basis_map","operation_detail":"A038243","conclusion_id":"n-ba196dcd01800741","conclusion_name":"A Jacobsthal triangle."}],"used_by_total":1,"links":[{"id":"l-2cc20928f31778fb","kind":"oeis_match","node_a_id":"n-54865b1c531f7155","node_b_id":null,"fingerprint":null,"oeis_id":"A143462","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A143462","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A143462","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-54865b1c531f7155","oeis_id":"A143462","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":30,"oeis_terms_available":30,"term_count":200,"raw_size":7803}],"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:     -5,      1;\n  2:     15,     -8,      1;\n  3:    -25,     35,    -11,      1;\n  4:    -25,    -90,     64,    -14,     1;\n  5:    375,     25,   -230,    102,   -17,     1;\n  6:  -1625,   1100,    355,   -472,   149,   -20,     1;\n  7:   4375,  -6625,   1675,   1235,  -843,   205,   -23,   1;\n  8:  -5625,  23250, -16350,    930,  3016, -1370,   270, -26,   1;\n  9: -15625, -46875,  75000, -30000, -3250,  6130, -2080, 344, -29, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(5-x)*P(n-1,x) - (10-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -5 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -5 + x;\nP[n_, x_] := P[n, x] = Expand[-(5 - x)*P[n - 1, x] - (10 - 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] = -5 + x; for(n = 2, N, my(i = n + 1); v[i] = -(5 - x)*v[i - 1] - (10 - 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\u22125"],"edges":["e-0c61b86c2e25257b"],"text":"A113953 \u2192 x \u21a6 x\u22125"},"headline":"A113953 \u2192 x \u21a6 x\u22125","chain":null}