{"id":"n-2de752bc9bc14cd4","revision":1,"first_index":1,"fingerprint":"f6-a6a00d92f59585d4","name":"shift_x_plus_one of Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-1","1"],["0","-2","1"],["1","1","-3","1"],["-1","2","3","-4","1"],["0","-4","2","6","-5","1"],["1","2","-9","0","10","-6","1"],["-1","3","9","-15","-5","15","-7","1"],["0","-6","3","24","-20","-14","21","-8","1"],["1","3","-18","-6","49","-21","-28","28","-9","1"],["-1","4","18","-36","-35","84","-14","-48","36","-10","1"],["0","-8","4","60","-50","-98","126","6","-75","45","-11","1"],["1","4","-30","-20","145","-36","-210","168","45","-110","55","-12","1"],["-1","5","30","-70","-115","279","48","-384","198","110","-154","66","-13","1"],["0","-10","5","120","-100","-358","441","264","-627","198","209","-208","78","-14","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","0","-1","0","1","0","-1","0","1","0","-1","0","1","0","-1"],"nonn":false,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["-1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","-2","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163659,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-1, 3)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"3"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 3)","region_exact":null,"edges":["e-7ddc15d309ad2a75"]},{"id":163660,"position":1,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-7ddc15d309ad2a75"]}],"region_exact":{},"incoming_edges":[{"id":"e-7ddc15d309ad2a75","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-2de752bc9bc14cd4","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":"3daff7e5b2fdadecd801855b63777a5bb1642e0ca5e64619982562ccb1902f3d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"3"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-f7e7f08472ec987a","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"0"},"upper":{"closed":false,"value":"4"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (0, 4)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-2af189ebe4d1038f","kind":"oeis_match","node_a_id":"n-2de752bc9bc14cd4","node_b_id":null,"fingerprint":null,"oeis_id":"A014757","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":15},"in_catalog":false,"oeis_id":"A014757","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-2e6d70f43098c75e","kind":"oeis_match","node_a_id":"n-2de752bc9bc14cd4","node_b_id":null,"fingerprint":null,"oeis_id":"A015021","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":15},"in_catalog":false,"oeis_id":"A015021","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-38f25e432657911d","kind":"oeis_match","node_a_id":"n-2de752bc9bc14cd4","node_b_id":null,"fingerprint":null,"oeis_id":"A014845","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":15},"in_catalog":false,"oeis_id":"A014845","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-45d2afcaf4f351a6","kind":"oeis_match","node_a_id":"n-2de752bc9bc14cd4","node_b_id":null,"fingerprint":null,"oeis_id":"A014581","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":15},"in_catalog":false,"oeis_id":"A014581","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-564679e180e2329a","kind":"oeis_match","node_a_id":"n-2de752bc9bc14cd4","node_b_id":null,"fingerprint":null,"oeis_id":"A011632","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":13},"in_catalog":false,"oeis_id":"A011632","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-621936dca5fc4e1a","kind":"oeis_match","node_a_id":"n-2de752bc9bc14cd4","node_b_id":null,"fingerprint":null,"oeis_id":"A014893","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":15},"in_catalog":false,"oeis_id":"A014893","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-8cd6bbb760ea8079","kind":"oeis_match","node_a_id":"n-2de752bc9bc14cd4","node_b_id":null,"fingerprint":null,"oeis_id":"A104562","in_catalog":0,"target":"triangle","row_alignment":"tabl","rows_matched":13,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":13,"target":"triangle","terms_compared":91},"in_catalog":false,"oeis_id":"A104562","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-e27292b3ca4ba3ae","kind":"oeis_match","node_a_id":"n-2de752bc9bc14cd4","node_b_id":null,"fingerprint":null,"oeis_id":"A015205","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":15},"in_catalog":false,"oeis_id":"A015205","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-fbdd2664416e3295","kind":"oeis_match","node_a_id":"n-2de752bc9bc14cd4","node_b_id":null,"fingerprint":null,"oeis_id":"A014997","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":15},"in_catalog":false,"oeis_id":"A014997","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A011632","relation":"match_row_sums"},{"oeis_id":"A014581","relation":"match_row_sums"},{"oeis_id":"A014757","relation":"match_row_sums"},{"oeis_id":"A014845","relation":"match_row_sums"},{"oeis_id":"A014893","relation":"match_row_sums"},{"oeis_id":"A014997","relation":"match_row_sums"},{"oeis_id":"A015021","relation":"match_row_sums"},{"oeis_id":"A015205","relation":"match_row_sums"},{"oeis_id":"A104562","relation":"match_triangle"}],"bfiles":[{"node_id":"n-2de752bc9bc14cd4","oeis_id":"A011632","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":13,"oeis_terms_agreeing":13,"oeis_terms_available":13,"term_count":200,"raw_size":1140},{"node_id":"n-2de752bc9bc14cd4","oeis_id":"A104562","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":91,"oeis_terms_agreeing":91,"oeis_terms_available":91,"term_count":990,"raw_size":10010}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2: -1,  1;\n   3:  0, -2,   1;\n   4:  1,  1,  -3,   1;\n   5: -1,  2,   3,  -4,   1;\n   6:  0, -4,   2,   6,  -5,   1;\n   7:  1,  2,  -9,   0,  10,  -6,   1;\n   8: -1,  3,   9, -15,  -5,  15,  -7,  1;\n   9:  0, -6,   3,  24, -20, -14,  21, -8,  1;\n  10:  1,  3, -18,  -6,  49, -21, -28, 28, -9, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = -(n-n*x)*P(n-1,x) - (3+2*x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - n*x)*P[n - 1, x] - (3 + 2*x - x^2)*D[P[n - 1, x], x])/(2*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(n - n*x)*v[i - 1] - (3 + 2*x - x^2)*deriv(v[i - 1], x))/(2*n - 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A104562","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1: -1,  1;\n  2:  0, -2,   1;\n  3:  1,  1,  -3,   1;\n  4: -1,  2,   3,  -4,   1;\n  5:  0, -4,   2,   6,  -5,   1;\n  6:  1,  2,  -9,   0,  10,  -6,   1;\n  7: -1,  3,   9, -15,  -5,  15,  -7,  1;\n  8:  0, -6,   3,  24, -20, -14,  21, -8,  1;\n  9:  1,  3, -18,  -6,  49, -21, -28, 28, -9, 1;\n","recurrence_sentence":"The row polynomials satisfy 2*n*P(n,x) = -(n+1-(n+1)*x)*P(n-1,x) - (3+2*x-x^2)*d/dx 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[(-(n + 1 - (n + 1)*x)*P[n - 1, x] - (3 + 2*x - x^2)*D[P[n - 1, x], x])/(2*n)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (-(n + 1 - (n + 1)*x)*v[i - 1] - (3 + 2*x - x^2)*deriv(v[i - 1], x))/(2*n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-f7e7f08472ec987a","steps":["x \u21a6 x+1"],"edges":["e-7ddc15d309ad2a75"],"text":"A053122 \u2192 x \u21a6 x+1"},"headline":"A053122 \u2192 x \u21a6 x+1","chain":null}
