{"id":"n-01fe968f3e05a927","revision":1,"first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","name":"Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","1"],["1","3","1"],["1","6","5","1"],["1","10","15","7","1"],["1","15","35","28","9","1"],["1","21","70","84","45","11","1"],["1","28","126","210","165","66","13","1"],["1","36","210","462","495","286","91","15","1"],["1","45","330","924","1287","1001","455","120","17","1"],["1","55","495","1716","3003","3003","1820","680","153","19","1"],["1","66","715","3003","6435","8008","6188","3060","969","190","21","1"],["1","78","1001","5005","12870","19448","18564","11628","4845","1330","231","23","1"],["1","91","1365","8008","24310","43758","50388","38760","20349","7315","1771","276","25","1"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","2","5","13","34","89","233","610","1597","4181","10946","28657","75025","196418"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A165253","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159477,"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-fe60348ba1bf754c"]},{"id":159478,"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-2580795600db03bb","e-365c9804d88a701d","e-fe60348ba1bf754c"]},{"id":159479,"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-2580795600db03bb"]},{"id":159480,"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-2580795600db03bb","e-365c9804d88a701d"]},{"id":159481,"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-2580795600db03bb","e-365c9804d88a701d"]},{"id":159482,"position":5,"claim":"real_rooted","from_index":0,"region_key":"(-4, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0)","region_exact":null,"edges":["e-365c9804d88a701d"]},{"id":159483,"position":6,"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-5ee7c33973be4058"]},{"id":159484,"position":7,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-1, 0]","output_interval":"(-inf, 0]","statement":"every PF input with all roots in [-1, 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 [-1, 0] to roots in (-inf, 0]: every PF input with all roots in [-1, 0] has a PF image; no larger [-R, 0] works","region_exact":null,"edges":["e-5ee7c33973be4058"]}],"region_exact":{},"incoming_edges":[{"id":"e-5ee7c33973be4058","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-01fe968f3e05a927","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":"67e1ce5ec01171ee7eb953367930989d71217a353511fd4eb728c7aa595b221d","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, 0]","output_interval":"(-inf, 0]","statement":"every PF input with all roots in [-1, 0] has a PF image; no larger [-R, 0] works"}],"premises":[]},{"id":"e-2580795600db03bb","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-01fe968f3e05a927","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":"c84a305c1d68c3e2d48bb6e3cce9a8b27585ac96fc10828306fe6273a8128d48","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},{"id":"e-365c9804d88a701d","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-01fe968f3e05a927","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":"e24029a8e3c32831442d0b04fa77751ddd1f2a9922c8c1688da77e95903b3570","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]},{"id":"e-fe60348ba1bf754c","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-01fe968f3e05a927","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":"89bbe0b232c6b35be8f33a53b4a761864393d79b754682eaf09d4330194a1f89","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}},{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-26a9c228139b7538","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-5d20c8ea29f3fbb9","conclusion_name":"basis_map_A008277 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938."},{"id":"e-4840ffa4ca7bbaf3","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-a18927332224e18f","conclusion_name":"basis_map_A013611 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938."},{"id":"e-66cd70bb24d83d57","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-1852b727a5d6b55d","conclusion_name":"basis_map_A013609 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938."},{"id":"e-c1b3c12b9e216f54","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-d0986a7acf60271a","conclusion_name":"shift_x_plus_one of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938."},{"id":"e-e3919df50aa70cdb","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-ee90405a2d071e0a","conclusion_name":"basis_map_A013610 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938."},{"id":"e-05447d659c882bb2","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-e420707ba29fb085","conclusion_name":"one_plus_derivative of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938."},{"id":"e-0a6146aa5a7bedf3","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-626f38256d467f88","conclusion_name":"theta_plus_one_squared of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938."},{"id":"e-4a7114805c9287a9","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-db31c2ee901d3a4d","conclusion_name":"theta_plus_one of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938."},{"id":"e-54cd73ac967e5a5b","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-dc3e6727bddaa3d6","conclusion_name":"derivative of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938."},{"id":"e-b3424874909a70f6","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-d0986a7acf60271a","conclusion_name":"shift_x_plus_one of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938."},{"id":"e-0aa38fea2ddc893c","operation":"veronese_section","operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-c017cea511485f1e","conclusion_name":"veronese_section_2_0 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938."},{"id":"e-af7162eb5065d83d","operation":"veronese_section","operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-e26bf661486fb5f5","conclusion_name":"veronese_section_2_1 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938."}],"used_by_total":12,"links":[{"id":"l-3a69cfb2d4ecbb42","kind":"fingerprint_related","node_a_id":"n-01fe968f3e05a927","node_b_id":"n-4c232e962100ae75","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-4c232e962100ae75","other":{"id":"n-4c232e962100ae75","name":"reversal of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","6","5","1"]],"row_sums_preview":["1","2","5","13","34","89","233","610"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6758fe7729845be8","steps":["reverse"],"edges":["e-99c1ee54068faff3"],"text":"A054142 \u2192 reverse"},"headline":"A054142 \u2192 reverse","chain":null}},{"id":"l-6edf36d7ad5e7467","kind":"fingerprint_related","node_a_id":"n-01fe968f3e05a927","node_b_id":"n-d8cfb1e39c5d56b0","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-d8cfb1e39c5d56b0","other":{"id":"n-d8cfb1e39c5d56b0","name":"reversal of Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","first_index":1,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","5","6","1"]],"row_sums_preview":["1","2","5","13","34","89","233","610"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0eba401b117f16a9","steps":["reverse"],"edges":["e-6116c33129f68e6c"],"text":"A085478 \u2192 reverse"},"headline":"A085478 \u2192 reverse","chain":null}},{"id":"l-709857531361d4c2","kind":"fingerprint_related","node_a_id":"n-01fe968f3e05a927","node_b_id":"n-ff36538314ab71f5","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-ff36538314ab71f5","other":{"id":"n-ff36538314ab71f5","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","4"],["16","48","16"]],"card_rows":[["1"],["4","4"],["16","48","16"],["64","320","384","64"]],"row_sums_preview":["1","8","80","832","8704","91136","954368","9994240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["basis A013611"],"edges":["e-36be6b9e5c42de00"],"text":"A103327 \u2192 basis A013611"},"headline":"A103327 \u2192 basis A013611","chain":null}},{"id":"l-8f3b19ff1e56e503","kind":"fingerprint_related","node_a_id":"n-6758fe7729845be8","node_b_id":"n-01fe968f3e05a927","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-6758fe7729845be8","other":{"id":"n-6758fe7729845be8","name":"Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","first_index":0,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","5","6","1"]],"row_sums_preview":["1","2","5","13","34","89","233","610"],"regions":["(-inf, +inf)","(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A054142"],"derivation":{"root":"n-6758fe7729845be8","steps":[],"edges":[],"text":"A054142"},"headline":"A054142: Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","chain":null}},{"id":"l-ecb0a26b0bfe6ede","kind":"fingerprint_related","node_a_id":"n-0eba401b117f16a9","node_b_id":"n-01fe968f3e05a927","fingerprint":"f6-2f8272d81c3bae1a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2f8272d81c3bae1a"},"other_id":"n-0eba401b117f16a9","other":{"id":"n-0eba401b117f16a9","name":"Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","first_index":1,"fingerprint":"f6-2f8272d81c3bae1a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","6","5","1"]],"row_sums_preview":["1","2","5","13","34","89","233","610"],"regions":["(-inf, +inf)","(-inf, 0)","(-4, 0)"],"region_exact":{},"oeis_ids":["A085478"],"derivation":{"root":"n-0eba401b117f16a9","steps":[],"edges":[],"text":"A085478"},"headline":"A085478: Triangle read by rows: T(n, k) = binomial(n + k, 2*k).","chain":null}},{"id":"l-1c8777978236b749","kind":"oeis_match","node_a_id":"n-01fe968f3e05a927","node_b_id":null,"fingerprint":null,"oeis_id":"A048575","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A048575","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-5ac65fb9d3fd3435","kind":"oeis_match","node_a_id":"n-01fe968f3e05a927","node_b_id":null,"fingerprint":null,"oeis_id":"A054142","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"reversed_triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A054142","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-91b57b11933f77c4","kind":"oeis_match","node_a_id":"n-01fe968f3e05a927","node_b_id":null,"fingerprint":null,"oeis_id":"A001519","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":"A001519","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c43aff6a531a40ea","kind":"oeis_match","node_a_id":"n-01fe968f3e05a927","node_b_id":null,"fingerprint":null,"oeis_id":"A085478","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A085478","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ebfb4a69deb245d7","kind":"oeis_match","node_a_id":"n-01fe968f3e05a927","node_b_id":null,"fingerprint":null,"oeis_id":"A122367","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":14},"in_catalog":false,"oeis_id":"A122367","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A165253","relation":"identification"},{"oeis_id":"A054142","relation":"match_reversed_triangle"},{"oeis_id":"A001519","relation":"match_row_sums"},{"oeis_id":"A048575","relation":"match_row_sums"},{"oeis_id":"A122367","relation":"match_row_sums"},{"oeis_id":"A085478","relation":"match_triangle"}],"bfiles":[{"node_id":"n-01fe968f3e05a927","oeis_id":"A001519","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":31,"oeis_terms_available":31,"term_count":200,"raw_size":9198},{"node_id":"n-01fe968f3e05a927","oeis_id":"A048575","target":"row_sums","oeis_offset":0,"first_row":2,"rows_used":200,"terms_matched":13,"oeis_terms_agreeing":29,"oeis_terms_available":29,"term_count":200,"raw_size":9364},{"node_id":"n-01fe968f3e05a927","oeis_id":"A054142","target":"reversed_triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":12893},{"node_id":"n-01fe968f3e05a927","oeis_id":"A085478","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":12893},{"node_id":"n-01fe968f3e05a927","oeis_id":"A122367","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":30,"oeis_terms_available":30,"term_count":200,"raw_size":9281}],"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;\n  2: 1,  1;\n  3: 1,  3,   1;\n  4: 1,  6,   5,   1;\n  5: 1, 10,  15,   7,   1;\n  6: 1, 15,  35,  28,   9,   1;\n  7: 1, 21,  70,  84,  45,  11,  1;\n  8: 1, 28, 126, 210, 165,  66, 13,  1;\n  9: 1, 36, 210, 462, 495, 286, 91, 15, 1;\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) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, 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] = 1; 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};\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":"A054142","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: 1,  1;\n  2: 1,  3,   1;\n  3: 1,  5,   6,   1;\n  4: 1,  7,  15,  10,    1;\n  5: 1,  9,  28,  35,   15,    1;\n  6: 1, 11,  45,  84,   70,   21,   1;\n  7: 1, 13,  66, 165,  210,  126,  28,   1;\n  8: 1, 15,  91, 286,  495,  462, 210,  36,  1;\n  9: 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1;\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) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 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] = 1 + 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}},{"oeis_id":"A085478","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: 1,  3,   1;\n  3: 1,  6,   5,   1;\n  4: 1, 10,  15,   7,    1;\n  5: 1, 15,  35,  28,    9,    1;\n  6: 1, 21,  70,  84,   45,   11,   1;\n  7: 1, 28, 126, 210,  165,   66,  13,   1;\n  8: 1, 36, 210, 462,  495,  286,  91,  15,  1;\n  9: 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1;\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) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 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] = 1 + 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}}],"derivation":{"root":"n-01fe968f3e05a927","steps":[],"edges":[],"text":"A165253"},"headline":"A165253: Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","chain":null}