{"id":"n-9b5c7e97e5191da6","revision":1,"first_index":0,"fingerprint":"f6-37c3559a5d3b5b73","name":"Triangular array formed by the Mersenne numbers.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","1"],["7","6","1"],["15","23","9","1"],["31","72","48","12","1"],["63","201","198","82","15","1"],["127","522","699","420","125","18","1"],["255","1291","2223","1795","765","177","21","1"],["511","3084","6562","6768","3840","1260","238","24","1"],["1023","7181","18324","23276","16758","7266","1932","308","27","1"],["2047","16398","49029","74616","65870","36036","12586","2808","387","30","1"],["4095","36879","126837","226325","238710","159446","69930","20394","3915","475","33","1"],["8191","81936","319332","656580","810715","644976","344064","125496","31365","5280","572","36","1"],["16383","180241","786258","1836422","2611305","2426751","1537308","679764","211761","46255","6930","678","39","1"],["32767","393234","1900351","4982364","8048907","8601606","6350547","3325608","1252317","339966","65901","8892","793","42","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","14","48","164","560","1912","6528","22288","76096","259808","887040","3028544","10340096","35303296"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A110441","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159045,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-7eb461f112fb8b45"]},{"id":159046,"position":1,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"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 >= 1","region_exact":null,"edges":["e-02d1e2cf79c791bd","e-7eb461f112fb8b45","e-c01294f1353c0463"]},{"id":159047,"position":2,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0)","region_exact":null,"edges":["e-c01294f1353c0463"]},{"id":159048,"position":3,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-02d1e2cf79c791bd","e-c01294f1353c0463"]},{"id":159049,"position":4,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-02d1e2cf79c791bd","e-c01294f1353c0463"]},{"id":159050,"position":5,"claim":"real_rooted","from_index":0,"region_key":"(-6, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-6"},"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-6, 0)","region_exact":null,"edges":["e-02d1e2cf79c791bd"]},{"id":159051,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-sqrt(2), sqrt(2)]","output_interval":"(-inf, +inf)","statement":"every real polynomial with all roots in E = [-sqrt(2), sqrt(2)] has a real-rooted image"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in [-sqrt(2), sqrt(2)] to roots in (-inf, +inf): every real polynomial with all roots in E = [-sqrt(2), sqrt(2)] has a real-rooted image","region_exact":null,"edges":["e-4036622c95205ccf"]},{"id":159052,"position":7,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-sqrt(2), sqrt(2)]","output_interval":"[-3 - 2 sqrt(2), -3 + 2 sqrt(2)]","statement":"every real polynomial with all roots in E = [-sqrt(2), sqrt(2)] has an image with all roots in I = [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in [-sqrt(2), sqrt(2)] to roots in [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]: every real polynomial with all roots in E = [-sqrt(2), sqrt(2)] has an image with all roots in I = [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]","region_exact":null,"edges":["e-4036622c95205ccf"]},{"id":159053,"position":8,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-sqrt(2), 0]","output_interval":"[-3 - 2 sqrt(2), -3 + 2 sqrt(2)]","statement":"every PF input with all roots in [-sqrt(2), 0] has a PF image; no larger [-R, 0] works"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in [-sqrt(2), 0] to roots in [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]: every PF input with all roots in [-sqrt(2), 0] has a PF image; no larger [-R, 0] works","region_exact":null,"edges":["e-4036622c95205ccf"]}],"region_exact":{},"incoming_edges":[{"id":"e-4036622c95205ccf","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-9b5c7e97e5191da6","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":"13b91352fa48e3929b531db58d4a916d99d0a2a0c37cf9f17fb7bbf3f6969bb8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"[-sqrt(2), sqrt(2)]","output_interval":"(-inf, +inf)","statement":"every real polynomial with all roots in E = [-sqrt(2), sqrt(2)] has a real-rooted image"},{"claim":"basis_map_preserver","input_interval":"[-sqrt(2), sqrt(2)]","output_interval":"[-3 - 2 sqrt(2), -3 + 2 sqrt(2)]","statement":"every real polynomial with all roots in E = [-sqrt(2), sqrt(2)] has an image with all roots in I = [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]"},{"claim":"basis_map_preserver","input_interval":"[-sqrt(2), 0]","output_interval":"[-3 - 2 sqrt(2), -3 + 2 sqrt(2)]","statement":"every PF input with all roots in [-sqrt(2), 0] has a PF image; no larger [-R, 0] works"}],"premises":[]},{"id":"e-02d1e2cf79c791bd","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-9b5c7e97e5191da6","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":"bf6a0d3c49c8fea9cc4b5ebf32a2fb24b2e813427fd329fa99ff7c81c490b4d8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-6"},"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},{"id":"e-7eb461f112fb8b45","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-9b5c7e97e5191da6","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":"c0ccb35358f13bef2dadb82246620f66a2531631b653fdc3d493f4acacf73569","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},{"id":"e-c01294f1353c0463","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-9b5c7e97e5191da6","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":"8d07a622bc0453c4a600c84abb91f2735b42857c5c03e772a33f7fc39858139e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-1d7f8f614531604f","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-6bdb5d7e28f56300","conclusion_name":"basis_map_A007318 of Triangular array formed by the Mersenne numbers."},{"id":"e-5de408358bde06b7","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-89d60963344cadbc","conclusion_name":"basis_map_A013610 of Triangular array formed by the Mersenne numbers."},{"id":"e-7622699fc8055eaa","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-5f5779bda3da22c9","conclusion_name":"basis_map_A008277 of Triangular array formed by the Mersenne numbers."},{"id":"e-94a352ca3242bd54","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-107a00b591d3c4c8","conclusion_name":"basis_map_A013609 of Triangular array formed by the Mersenne numbers."},{"id":"e-b936d8f2208fd69d","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-35273d9764d27bc4","conclusion_name":"basis_map_A013611 of Triangular array formed by the Mersenne numbers."},{"id":"e-5bec3e19f7ad29fa","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-6bdb5d7e28f56300","conclusion_name":"basis_map_A007318 of Triangular array formed by the Mersenne numbers."},{"id":"e-5faf1c18373ba45e","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-369f42b9099fbd84","conclusion_name":"reversal of Triangular array formed by the Mersenne numbers."},{"id":"e-7ce85da28c41d2d5","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-ee0097aa03dd5d36","conclusion_name":"derivative of Triangular array formed by the Mersenne numbers."},{"id":"e-813d9e0ebd3ed0a7","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-cf288f0202bf1453","conclusion_name":"mobius_x_over_x_plus_one of Triangular array formed by the Mersenne numbers."},{"id":"e-9b8106d62ea669b5","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-e70bc71f808d87be","conclusion_name":"one_plus_derivative of Triangular array formed by the Mersenne numbers."},{"id":"e-ec8938d063ea19e6","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-d391e9db43dc4133","conclusion_name":"theta_plus_one of Triangular array formed by the Mersenne numbers."},{"id":"e-f01a2d79bc5e0611","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8d7b2a6627d99246","conclusion_name":"theta_plus_one_squared of Triangular array formed by the Mersenne numbers."},{"id":"e-91081c80c4ccf2eb","operation":"veronese_section","operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-51911d2f3d09d8ee","conclusion_name":"veronese_section_2_1 of Triangular array formed by the Mersenne numbers."},{"id":"e-963a8b66c751c67d","operation":"veronese_section","operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-c141c207f11d8598","conclusion_name":"veronese_section_2_0 of Triangular array formed by the Mersenne numbers."}],"used_by_total":14,"links":[{"id":"l-1a71ed9a7dc2f579","kind":"fingerprint_related","node_a_id":"n-9b5c7e97e5191da6","node_b_id":"n-369f42b9099fbd84","fingerprint":"f6-37c3559a5d3b5b73","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-37c3559a5d3b5b73"},"other_id":"n-369f42b9099fbd84","other":{"id":"n-369f42b9099fbd84","name":"reversal of Triangular array formed by the Mersenne numbers.","first_index":0,"fingerprint":"f6-37c3559a5d3b5b73","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","6","7"]],"card_rows":[["1"],["1","3"],["1","6","7"],["1","9","23","15"]],"row_sums_preview":["1","4","14","48","164","560","1912","6528"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9b5c7e97e5191da6","steps":["reverse"],"edges":["e-5faf1c18373ba45e"],"text":"A110441 \u2192 reverse"},"headline":"A110441 \u2192 reverse","chain":null}},{"id":"l-87405bc63b6d87b4","kind":"oeis_match","node_a_id":"n-9b5c7e97e5191da6","node_b_id":null,"fingerprint":null,"oeis_id":"A204089","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"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":"A204089","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b89915a73fe736ad","kind":"oeis_match","node_a_id":"n-9b5c7e97e5191da6","node_b_id":null,"fingerprint":null,"oeis_id":"A110441","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A110441","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ec84b7b59214ea54","kind":"oeis_match","node_a_id":"n-9b5c7e97e5191da6","node_b_id":null,"fingerprint":null,"oeis_id":"A007070","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"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":"A007070","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A110441","relation":"identification"},{"oeis_id":"A007070","relation":"match_row_sums"},{"oeis_id":"A204089","relation":"match_row_sums"},{"oeis_id":"A110441","relation":"match_triangle"}],"bfiles":[{"node_id":"n-9b5c7e97e5191da6","oeis_id":"A007070","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":28,"oeis_terms_available":28,"term_count":200,"raw_size":11624},{"node_id":"n-9b5c7e97e5191da6","oeis_id":"A110441","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":16076}],"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:    3,    1;\n  2:    7,    6,     1;\n  3:   15,   23,     9,     1;\n  4:   31,   72,    48,    12,     1;\n  5:   63,  201,   198,    82,    15,    1;\n  6:  127,  522,   699,   420,   125,   18,    1;\n  7:  255, 1291,  2223,  1795,   765,  177,   21,   1;\n  8:  511, 3084,  6562,  6768,  3840, 1260,  238,  24,  1;\n  9: 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) - 2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] - 2*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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (3 + x)*v[i - 1] - 2*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}},{"oeis_id":"A110441","target":"triangle","first_row":0,"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:    3,    1;\n  2:    7,    6,     1;\n  3:   15,   23,     9,     1;\n  4:   31,   72,    48,    12,     1;\n  5:   63,  201,   198,    82,    15,    1;\n  6:  127,  522,   699,   420,   125,   18,    1;\n  7:  255, 1291,  2223,  1795,   765,  177,   21,   1;\n  8:  511, 3084,  6562,  6768,  3840, 1260,  238,  24,  1;\n  9: 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+x)*P(n-1,x) - 2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(3 + x)*P[n - 1, x] - 2*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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (3 + x)*v[i - 1] - 2*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":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-9b5c7e97e5191da6","steps":[],"edges":[],"text":"A110441"},"headline":"A110441: Triangular array formed by the Mersenne numbers.","chain":null}
