{"id":"n-87e7c3f73bb81aac","revision":1,"first_index":0,"fingerprint":"f6-560ad1ce30eafb0f","name":"basis-map coefficients of Riordan array (1+2x,x(1+x)).","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"],["-1","1"],["0","-3","1"],["3","5","-5","1"],["-9","-3","14","-7","1"],["18","-12","-25","27","-9","1"],["-27","54","18","-71","44","-11","1"],["27","-135","63","125","-149","65","-13","1"],["0","243","-324","-81","369","-267","90","-15","1"],["-81","-297","891","-393","-616","826","-433","119","-17","1"],["243","81","-1755","1989","267","-1924","1588","-655","152","-19","1"],["-486","810","2376","-5652","2643","2945","-4563","2763","-941","189","-21","1"],["729","-2916","-972","11610","-12393","-153","9946","-9299","4475","-1299","230","-23","1"],["-729","6561","-6318","-16470","35208","-18126","-13357","24991","-17138","6864","-1737","275","-25","1"],["0","-10935","25515","7290","-73305","77652","-8046","-50487","53681","-29358","10086","-2263","324","-27","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","0","-2","4","-4","0","8","-16","16","0","-32","64","-64","0","128"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_coefficients_of:oeis_catalog:A113214","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["-1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160548,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-1, 3]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":true,"value":"3"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-1, 3]","region_exact":null,"edges":["e-f73e26dac50fbf3d"]},{"id":160549,"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-f73e26dac50fbf3d"]}],"region_exact":{},"incoming_edges":[{"id":"e-f73e26dac50fbf3d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-3\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-87e7c3f73bb81aac","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":"20f81b317223a2aab5566929da949684cb75ed7ceb965832896e2915f1367f46","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":true,"value":"3"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c27b11147fce8bfa","role":"sequence","name":"Riordan array (1+2x,x(1+x)).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]"}]}],"used_by":[{"id":"e-f9e5e2bc826d3dc0","operation":"basis_map","operation_detail":"A027465","conclusion_id":"n-c27b11147fce8bfa","conclusion_name":"Riordan array (1+2x,x(1+x))."}],"used_by_total":1,"links":[{"id":"l-ed28ee254638d4a0","kind":"oeis_match","node_a_id":"n-87e7c3f73bb81aac","node_b_id":null,"fingerprint":null,"oeis_id":"A090132","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":"A090132","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A090132","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-87e7c3f73bb81aac","oeis_id":"A090132","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":47,"oeis_terms_available":47,"term_count":200,"raw_size":3350}],"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:  -1,    1;\n  2:   0,   -3,    1;\n  3:   3,    5,   -5,    1;\n  4:  -9,   -3,   14,   -7,    1;\n  5:  18,  -12,  -25,   27,   -9,    1;\n  6: -27,   54,   18,  -71,   44,  -11,    1;\n  7:  27, -135,   63,  125, -149,   65,  -13,   1;\n  8:   0,  243, -324,  -81,  369, -267,   90, -15,   1;\n  9: -81, -297,  891, -393, -616,  826, -433, 119, -17, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3-x)*P(n-1,x) - (3-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -1 + x;\nP[n_, x_] := P[n, x] = Expand[-(3 - x)*P[n - 1, x] - (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] = -1 + x; for(n = 2, N, my(i = n + 1); v[i] = -(3 - x)*v[i - 1] - (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-c27b11147fce8bfa","steps":["x \u21a6 x\u22123"],"edges":["e-f73e26dac50fbf3d"],"text":"A113214 \u2192 x \u21a6 x\u22123"},"headline":"A113214 \u2192 x \u21a6 x\u22123","chain":null}