{"id":"n-81eeb524d0837717","revision":1,"first_index":0,"fingerprint":"f6-9f0f04d1052f691c","name":"basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).","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"],["3","-4","1"],["-3","10","-6","1"],["0","-18","21","-8","1"],["9","21","-53","36","-10","1"],["-27","0","99","-116","55","-12","1"],["54","-81","-117","286","-215","78","-14","1"],["-81","270","-27","-528","650","-358","105","-16","1"],["81","-594","621","582","-1547","1275","-553","136","-18","1"],["0","972","-2106","432","2745","-3648","2261","-808","171","-20","1"],["-243","-1053","4833","-4527","-2580","8317","-7497","3724","-1131","210","-22","1"],["729","0","-8262","15012","-4590","-13842","20377","-13984","5796","-1530","253","-24","1"],["-1458","3645","9234","-34884","31995","9405","-44165","43660","-24255","8625","-2013","300","-26","1"],["2187","-11664","729","60588","-102087","40716","66927","-112816","85053","-39744","12375","-2588","351","-28","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-1","0","2","-4","4","0","-8","16","-16","0","32","-64","64","0"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_coefficients_of:oeis_catalog:A030528","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["-2","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":160014,"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-a2e7017ebd9e3749"]},{"id":160015,"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-a2e7017ebd9e3749"]}],"region_exact":{},"incoming_edges":[{"id":"e-a2e7017ebd9e3749","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-3\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-81eeb524d0837717","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":"9a3ebf5ddfef7542ee356144ece4a7904de21a76e5cf2837bee36330b9661929","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-21854f4257effb01","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(k,n-k).","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-5c2b8bc6b43c730e","operation":"basis_map","operation_detail":"A027465","conclusion_id":"n-21854f4257effb01","conclusion_name":"Triangle read by rows: T(n,k) = binomial(k,n-k)."}],"used_by_total":1,"links":[{"id":"l-0d389e427afca334","kind":"oeis_match","node_a_id":"n-81eeb524d0837717","node_b_id":null,"fingerprint":null,"oeis_id":"A009116","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":"A009116","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009116","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-81eeb524d0837717","oeis_id":"A009116","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":200,"raw_size":3320}],"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:   3,   -4,    1;\n  3:  -3,   10,   -6,    1;\n  4:   0,  -18,   21,   -8,     1;\n  5:   9,   21,  -53,   36,   -10,    1;\n  6: -27,    0,   99, -116,    55,  -12,    1;\n  7:  54,  -81, -117,  286,  -215,   78,  -14,   1;\n  8: -81,  270,  -27, -528,   650, -358,  105, -16,   1;\n  9:  81, -594,  621,  582, -1547, 1275, -553, 136, -18, 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) = -2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -2 + 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] = -2 + 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-21854f4257effb01","steps":["x \u21a6 x\u22123"],"edges":["e-a2e7017ebd9e3749"],"text":"A030528 \u2192 x \u21a6 x\u22123"},"headline":"A030528 \u2192 x \u21a6 x\u22123","chain":null}