{"id":"n-8b17fbdec05b5547","revision":0,"first_index":0,"fingerprint":"f6-8d9e8ddc34a45dd2","name":"Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["6","4"],["24","18","1"],["120","96","9"],["720","600","72","1"],["5040","4320","600","16"],["40320","35280","5400","200","1"],["362880","322560","52920","2400","25"],["3628800","3265920","564480","29400","450","1"],["39916800","36288000","6531840","376320","7350","36"],["479001600","439084800","81648000","5080320","117600","882","1"],["6227020800","5748019200","1097712000","72576000","1905120","18816","49"],["87178291200","80951270400","15807052800","1097712000","31752000","381024","1568","1"],["1307674368000","1220496076800","242853811200","17563392000","548856000","7620480","42336","64"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["1","3","10","43","225","1393","9976","81201","740785","7489051","83120346","1004933203","13147251985","185066460993","2789144166880"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A084950","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"],["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169198,"position":0,"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-9237a107fcd4a149"]},{"id":169199,"position":1,"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-9237a107fcd4a149"]},{"id":169200,"position":2,"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-9237a107fcd4a149"]},{"id":169201,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-9237a107fcd4a149"]}],"region_exact":{},"incoming_edges":[{"id":"e-9237a107fcd4a149","operation":{"operation":"theorem_check","theorem":"liu_wang_three_term"},"operation_detail":"liu_wang_three_term","conclusion_id":"n-8b17fbdec05b5547","premise_count":0,"theorem":"L. L. Liu and Y. Wang, Adv. Appl. Math. 38 (2007) 542-560, Theorem 2.1 and Corollary 2.4 (A unified approach to polynomial sequences with only real zeros; the lag row as the interlacer, b_n = 0), weak interlacing of consecutive rows, with nonnegativity of the coefficients replaced, where a root region is certified, by an exact root-location induction","lean_route":null,"lean_declarations":["RealRooted.MaWang.strictInterl_of_interlaces_evalCoeff_nonpos"],"certificate_hash":"795b35a7a36621cff00b3fc91a4026156dd7b1317d5a9ac0116453d4f67682ea","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":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-18088908dd2db715","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-f9ef7875a11a5ecd","conclusion_name":"shift_x_plus_one of Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients."},{"id":"e-a328da612d1a3acb","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-8f6d2c8dc196f9ed","conclusion_name":"basis_map_A013610 of Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients."},{"id":"e-d8af37fe2cd1c14d","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-8a9d356b74002a5f","conclusion_name":"basis_map_A013609 of Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients."},{"id":"e-e8247e43263f967d","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-fabff891e62e7553","conclusion_name":"basis_map_A013611 of Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients."},{"id":"e-e9f5665df89a8ea4","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-79f84b3c956053a7","conclusion_name":"basis_map_A008277 of Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients."},{"id":"e-1e21898189db93af","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-f9ef7875a11a5ecd","conclusion_name":"shift_x_plus_one of Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients."},{"id":"e-2fa13fafcaee11c4","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-3c734dc1c0de1b65","conclusion_name":"derivative of Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients."},{"id":"e-58421abae90fba29","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-e6cef00316b8374e","conclusion_name":"theta_plus_one of Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients."},{"id":"e-bfa4e89f4a7239cd","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-926cc35d1e5131de","conclusion_name":"theta_plus_one_squared of Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients."},{"id":"e-f35d919a2d6dbcdd","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-78b9be25caf5ba66","conclusion_name":"one_plus_derivative of Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients."}],"used_by_total":10,"links":[{"id":"l-b464728b2b56ae44","kind":"oeis_match","node_a_id":"n-8b17fbdec05b5547","node_b_id":null,"fingerprint":null,"oeis_id":"A001040","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001040","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A084950","relation":"identification"},{"oeis_id":"A001040","relation":"match_row_sums"}],"bfiles":[],"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:       6,       4;\n  3:      24,      18,      1;\n  4:     120,      96,      9;\n  5:     720,     600,     72,     1;\n  6:    5040,    4320,    600,    16;\n  7:   40320,   35280,   5400,   200,   1;\n  8:  362880,  322560,  52920,  2400,  25;\n  9: 3628800, 3265920, 564480, 29400, 450, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+1)*P(n-1,x) + 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[(n + 1)*P[n - 1, x] + 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] = (n + 1)*v[i - 1] + 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-8b17fbdec05b5547","steps":[],"edges":[],"text":"A084950"},"headline":"A084950: Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients.","chain":null}
