{"id":"n-366c29e67ccc8d94","revision":1,"first_index":0,"fingerprint":"f6-348de79756a0b070","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.","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"],["-2","0","1"],["-1","-3","1","1"],["1","-4","-3","2","1"],["2","0","-8","-2","3","1"],["1","6","-5","-12","0","4","1"],["-1","7","9","-15","-15","3","5","1"],["-2","0","21","6","-30","-16","7","6","1"],["-1","-9","12","42","-9","-49","-14","12","7","1"],["1","-10","-18","48","63","-42","-70","-8","18","8","1"],["2","0","-40","-12","120","70","-98","-90","3","25","9","1"],["1","12","-22","-100","45","232","42","-180","-105","20","33","10","1"],["-1","13","30","-110","-175","207","372","-48","-288","-110","44","42","11","1"],["-2","0","65","20","-330","-200","537","504","-231","-418","-99","76","52","12","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","0","-1","-2","-3","-4","-5","-6","-7","-8","-9","-10","-11","-12","-13"],"nonn":false,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2"],["-8"],["8"]]},"terms":[{"coeff":{"coeffs":[["1","0"],["-10","-2"],["7","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","-13","-2"],["-12","8","4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["9","-6","-3"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160883,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-3, 2)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"2"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-3, 2)","region_exact":null,"edges":["e-480f02718c8257df"]},{"id":160884,"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-480f02718c8257df"]}],"region_exact":{},"incoming_edges":[{"id":"e-480f02718c8257df","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-366c29e67ccc8d94","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":"72ea3ab20ae1f1f76b3282a701bab463c2cfb900777726d269fca2f8df0d15ce","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"2"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4ca7e225b5e9bcc6","role":"sequence","name":"Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-2"},"upper":{"closed":false,"value":"3"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 3)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-0f38b19605f6c689","kind":"oeis_match","node_a_id":"n-366c29e67ccc8d94","node_b_id":null,"fingerprint":null,"oeis_id":"A001489","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A001489","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-5874e29af5c72b52","kind":"oeis_match","node_a_id":"n-366c29e67ccc8d94","node_b_id":null,"fingerprint":null,"oeis_id":"A022959","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A022959","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-7bb7b25d0656cd04","kind":"oeis_match","node_a_id":"n-366c29e67ccc8d94","node_b_id":null,"fingerprint":null,"oeis_id":"A001478","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A001478","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-c01ccaf6321576b7","kind":"oeis_match","node_a_id":"n-366c29e67ccc8d94","node_b_id":null,"fingerprint":null,"oeis_id":"A024000","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":"A024000","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-e250f4e9ee81abd4","kind":"oeis_match","node_a_id":"n-366c29e67ccc8d94","node_b_id":null,"fingerprint":null,"oeis_id":"A022958","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":"A022958","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-fa1e2ab3c2825821","kind":"oeis_match","node_a_id":"n-366c29e67ccc8d94","node_b_id":null,"fingerprint":null,"oeis_id":"A022960","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A022960","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001478","relation":"match_row_sums"},{"oeis_id":"A001489","relation":"match_row_sums"},{"oeis_id":"A022958","relation":"match_row_sums"},{"oeis_id":"A022959","relation":"match_row_sums"},{"oeis_id":"A022960","relation":"match_row_sums"},{"oeis_id":"A024000","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-366c29e67ccc8d94","oeis_id":"A001478","target":"row_sums","oeis_offset":1,"first_row":2,"rows_used":200,"terms_matched":13,"oeis_terms_agreeing":65,"oeis_terms_available":65,"term_count":200,"raw_size":1584},{"node_id":"n-366c29e67ccc8d94","oeis_id":"A001489","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":200,"raw_size":1579},{"node_id":"n-366c29e67ccc8d94","oeis_id":"A024000","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":58,"oeis_terms_available":58,"term_count":200,"raw_size":1576}],"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: -2,  0,  1;\n  3: -1, -3,  1,   1;\n  4:  1, -4, -3,   2,   1;\n  5:  2,  0, -8,  -2,   3,   1;\n  6:  1,  6, -5, -12,   0,   4,   1;\n  7: -1,  7,  9, -15, -15,   3,   5,  1;\n  8: -2,  0, 21,   6, -30, -16,   7,  6, 1;\n  9: -1, -9, 12,  42,  -9, -49, -14, 12, 7, 1;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-8*n+2)*P(n,x) = (7*n^2-10*n+1+(4*n^2-2*n)*x)*P(n-1,x) - (12*n+3-(8*n-13)*x-(4*n-2)*x^2)*d/dx P(n-1,x) + (9-6*x-3*x^2)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((7*n^2 - 10*n + 1 + (4*n^2 - 2*n)*x)*P[n - 1, x] - (12*n + 3 - (8*n - 13)*x - (4*n - 2)*x^2)*D[P[n - 1, x], x] + (9 - 6*x - 3*x^2)*D[P[n - 1, x], {x, 2}])/(8*n^2 - 8*n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((7*n^2 - 10*n + 1 + (4*n^2 - 2*n)*x)*v[i - 1] - (12*n + 3 - (8*n - 13)*x - (4*n - 2)*x^2)*deriv(v[i - 1], x) + (9 - 6*x - 3*x^2)*deriv(deriv(v[i - 1], x), x))/(8*n^2 - 8*n + 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-4ca7e225b5e9bcc6","steps":["x \u21a6 x+1"],"edges":["e-480f02718c8257df"],"text":"A124038 \u2192 x \u21a6 x+1"},"headline":"A124038 \u2192 x \u21a6 x+1","chain":null}
