{"id":"n-a2d7d802498a9d22","revision":1,"first_index":1,"fingerprint":"f6-def41fdfbf381d0a","name":"Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents 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"],["0","2"],["-1","0","4"],["0","-4","0","8"],["1","0","-12","0","16"],["0","6","0","-32","0","32"],["-1","0","24","0","-80","0","64"],["0","-8","0","80","0","-192","0","128"],["1","0","-40","0","240","0","-448","0","256"],["0","10","0","-160","0","672","0","-1024","0","512"],["-1","0","60","0","-560","0","1792","0","-2304","0","1024"],["0","-12","0","280","0","-1792","0","4608","0","-5120","0","2048"],["1","0","-84","0","1120","0","-5376","0","11520","0","-11264","0","4096"],["0","14","0","-448","0","4032","0","-15360","0","28160","0","-24576","0","8192"],["-1","0","112","0","-2016","0","13440","0","-42240","0","67584","0","-53248","0","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15"],"nonn":false,"definitions":[{"position":0,"source":"oeis_catalog:A053117","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["0","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158729,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-b99a39d00579baec"]},{"id":158730,"position":1,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2","region_exact":null,"edges":["e-b99a39d00579baec","e-dc563681f97c05c6"]},{"id":158731,"position":2,"claim":"real_rooted","from_index":1,"region_key":"(-1, 1)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-1, 1)","region_exact":null,"edges":["e-dc563681f97c05c6"]},{"id":158732,"position":3,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-1, 1]","output_interval":"(-inf, +inf)","statement":"every real polynomial with all roots in E = [-1, 1] has a real-rooted image"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in [-1, 1] to roots in (-inf, +inf): every real polynomial with all roots in E = [-1, 1] has a real-rooted image","region_exact":null,"edges":["e-85cb7c7abd19f98e"]},{"id":158733,"position":4,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-1, 1]","output_interval":"[-1, 1]","statement":"every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [-1, 1]"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in [-1, 1] to roots in [-1, 1]: every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [-1, 1]","region_exact":null,"edges":["e-85cb7c7abd19f98e"]}],"region_exact":{},"incoming_edges":[{"id":"e-85cb7c7abd19f98e","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-a2d7d802498a9d22","premise_count":0,"theorem":"affine current multiplier with an affine initial row (module documentation, \"The affine-current theorem\"): with beta = b/d, K = beta^2 + a beta - c > 0 and eta = m/d >= 0, every real polynomial with all roots in E = [-beta - sqrt(K), -beta + sqrt(K)] has a real-rooted image, and no interval of positive length outside E works (Theorem 1); for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"2f85d5aec1776b399fac69ff8b62435e92eaf37f45748bd17b7ad89ee835f4c0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"[-1, 1]","output_interval":"(-inf, +inf)","statement":"every real polynomial with all roots in E = [-1, 1] has a real-rooted image"},{"claim":"basis_map_preserver","input_interval":"[-1, 1]","output_interval":"[-1, 1]","statement":"every real polynomial with all roots in E = [-1, 1] has an image with all roots in I = [-1, 1]"}],"premises":[]},{"id":"e-b99a39d00579baec","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-a2d7d802498a9d22","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"9c8c0f712b353f9ad179130425653ff88a5d15ca1fedb284e7bd19f6f9b7e7d2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}},{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},{"id":"e-dc563681f97c05c6","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-a2d7d802498a9d22","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"500bfedf127efcc72f599eec6b491175d096a5fcc7dc3172a81b97462b825cf6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":false,"value":"1"}}},{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-5342624c2722e8d8","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-b1a7e7698038fd9e","conclusion_name":"shift_x_plus_one of Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order)."},{"id":"e-aa85a20796e296f5","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-776f7a479f11ea52","conclusion_name":"derivative of Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order)."},{"id":"e-df5dd69f4658c66c","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-1654fe21a4718e1d","conclusion_name":"one_plus_derivative of Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order)."}],"used_by_total":3,"links":[{"id":"l-13989813831668af","kind":"fingerprint_related","node_a_id":"n-a2d7d802498a9d22","node_b_id":"n-02ad05e8959bc17f","fingerprint":"f6-def41fdfbf381d0a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-def41fdfbf381d0a"},"other_id":"n-02ad05e8959bc17f","other":{"id":"n-02ad05e8959bc17f","name":"derivative of Triangle of coefficients of cos(x)^n in polynomial for cos(nx).","first_index":0,"fingerprint":"f6-def41fdfbf381d0a","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[[],[],["0","-2"]],"card_rows":[["0","-2"],["0","-6"],["0","-16","0","4"],["0","-40","0","20"]],"row_sums_preview":["0","0","-2","-6","-12","-20","-30","-42"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dbf182f3ecfeda2a","steps":["D"],"edges":["e-6f899a70e4c27d14"],"text":"A039991 \u2192 D"},"headline":"A039991 \u2192 D","chain":null}},{"id":"l-cdbe3d4d8ac3c861","kind":"fingerprint_related","node_a_id":"n-ef9f4d5289b8feff","node_b_id":"n-a2d7d802498a9d22","fingerprint":"f6-def41fdfbf381d0a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-def41fdfbf381d0a"},"other_id":"n-ef9f4d5289b8feff","other":{"id":"n-ef9f4d5289b8feff","name":"chebyshev_u","first_index":0,"fingerprint":"f6-def41fdfbf381d0a","polytool_family":"chebyshev_u","degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["0","2"],["-1","0","4"]],"card_rows":[["1"],["0","2"],["-1","0","4"],["0","-4","0","8"]],"row_sums_preview":["1","2","3","4","5","6","7","8"],"regions":["(-inf, +inf)","(-1, 1)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ef9f4d5289b8feff","steps":[],"edges":[],"text":"chebyshev_u"},"headline":"chebyshev_u","chain":null}},{"id":"l-efa7b2c2721fea50","kind":"fingerprint_related","node_a_id":"n-a2d7d802498a9d22","node_b_id":"n-6ee7e6c62309a510","fingerprint":"f6-def41fdfbf381d0a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-def41fdfbf381d0a"},"other_id":"n-6ee7e6c62309a510","other":{"id":"n-6ee7e6c62309a510","name":"derivative of chebyshev_t","first_index":0,"fingerprint":"f6-def41fdfbf381d0a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[[],["1"],["0","4"]],"card_rows":[["1"],["0","4"],["-3","0","12"],["0","-16","0","32"]],"row_sums_preview":["0","1","4","9","16","25","36","49"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9ae8d431f7b33f22","steps":["D"],"edges":["e-c1cd3b615621d834"],"text":"chebyshev_t \u2192 D"},"headline":"chebyshev_t \u2192 D","chain":null}},{"id":"l-00944a2437670618","kind":"oeis_match","node_a_id":"n-a2d7d802498a9d22","node_b_id":null,"fingerprint":null,"oeis_id":"A004830","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A004830","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-062bdd7dcfe186a1","kind":"oeis_match","node_a_id":"n-a2d7d802498a9d22","node_b_id":null,"fingerprint":null,"oeis_id":"A002837","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A002837","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-0a906ceea2ac3757","kind":"oeis_match","node_a_id":"n-a2d7d802498a9d22","node_b_id":null,"fingerprint":null,"oeis_id":"A004438","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A004438","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-1e6ff89d80be1ac3","kind":"oeis_match","node_a_id":"n-a2d7d802498a9d22","node_b_id":null,"fingerprint":null,"oeis_id":"A000027","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000027","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-44cf0ad560c5e2ef","kind":"oeis_match","node_a_id":"n-a2d7d802498a9d22","node_b_id":null,"fingerprint":null,"oeis_id":"A001477","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001477","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9b8ade5e407e5c11","kind":"oeis_match","node_a_id":"n-a2d7d802498a9d22","node_b_id":null,"fingerprint":null,"oeis_id":"A053118","in_catalog":0,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":12,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":12,"target":"reversed_triangle","terms_compared":85},"in_catalog":false,"oeis_id":"A053118","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-a2ff5f0305ae9f23","kind":"oeis_match","node_a_id":"n-a2d7d802498a9d22","node_b_id":null,"fingerprint":null,"oeis_id":"A005496","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"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":"A005496","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c4bbbe667238c23b","kind":"oeis_match","node_a_id":"n-a2d7d802498a9d22","node_b_id":null,"fingerprint":null,"oeis_id":"A004441","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A004441","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f9fea37a78e2f1a2","kind":"oeis_match","node_a_id":"n-a2d7d802498a9d22","node_b_id":null,"fingerprint":null,"oeis_id":"A053117","in_catalog":1,"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":true,"oeis_id":"A053117","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-fbb50cfb44abbec1","kind":"oeis_match","node_a_id":"n-a2d7d802498a9d22","node_b_id":null,"fingerprint":null,"oeis_id":"A002377","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A002377","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A053117","relation":"identification"},{"oeis_id":"A053118","relation":"match_reversed_triangle"},{"oeis_id":"A000027","relation":"match_row_sums"},{"oeis_id":"A001477","relation":"match_row_sums"},{"oeis_id":"A002377","relation":"match_row_sums"},{"oeis_id":"A002837","relation":"match_row_sums"},{"oeis_id":"A004438","relation":"match_row_sums"},{"oeis_id":"A004441","relation":"match_row_sums"},{"oeis_id":"A004830","relation":"match_row_sums"},{"oeis_id":"A005496","relation":"match_row_sums"},{"oeis_id":"A053117","relation":"match_triangle"}],"bfiles":[{"node_id":"n-a2d7d802498a9d22","oeis_id":"A000027","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":77,"oeis_terms_available":77,"term_count":200,"raw_size":1384},{"node_id":"n-a2d7d802498a9d22","oeis_id":"A053117","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":9808},{"node_id":"n-a2d7d802498a9d22","oeis_id":"A053118","target":"reversed_triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":85,"oeis_terms_agreeing":85,"oeis_terms_available":85,"term_count":990,"raw_size":9808}],"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:  0,  2;\n   3: -1,  0,   4;\n   4:  0, -4,   0,    8;\n   5:  1,  0, -12,    0,  16;\n   6:  0,  6,   0,  -32,   0,   32;\n   7: -1,  0,  24,    0, -80,    0,   64;\n   8:  0, -8,   0,   80,   0, -192,    0,   128;\n   9:  1,  0, -40,    0, 240,    0, -448,     0, 256;\n  10:  0, 10,   0, -160,   0,  672,    0, -1024,   0, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) - P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] - P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = 2*x*v[i - 1] - v[i - 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":"A053117","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:  0,  2;\n  2: -1,  0,   4;\n  3:  0, -4,   0,    8;\n  4:  1,  0, -12,    0,  16;\n  5:  0,  6,   0,  -32,   0,   32;\n  6: -1,  0,  24,    0, -80,    0,   64;\n  7:  0, -8,   0,   80,   0, -192,    0,   128;\n  8:  1,  0, -40,    0, 240,    0, -448,     0, 256;\n  9:  0, 10,   0, -160,   0,  672,    0, -1024,   0, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*x*P(n-1,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[2*x*P[n - 1, x] - P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = 2*x*v[i - 1] - v[i - 2]); 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}},{"oeis_id":"A053118","target":"reversed_triangle","first_row":1,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   2, 0;\n  2:   4, 0,    -1;\n  3:   8, 0,    -4, 0;\n  4:  16, 0,   -12, 0,   1;\n  5:  32, 0,   -32, 0,   6, 0;\n  6:  64, 0,   -80, 0,  24, 0,   -1;\n  7: 128, 0,  -192, 0,  80, 0,   -8, 0;\n  8: 256, 0,  -448, 0, 240, 0,  -40, 0,  1;\n  9: 512, 0, -1024, 0, 672, 0, -160, 0, 10, 0;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = 2*x*P(n-1,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[2*x*P[n - 1, x] - P[n - 2, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = 2*x*v[i - 1] - v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-a2d7d802498a9d22","steps":[],"edges":[],"text":"A053117"},"headline":"A053117: Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).","chain":null}