{"id":"n-14dfd91463cae288","revision":1,"first_index":0,"fingerprint":"f6-867f73b103ecf726","name":"Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).","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"],["1","3"],["1","5","3"],["1","7","11","3"],["1","9","23","17","3"],["1","11","39","51","23","3"],["1","13","59","113","91","29","3"],["1","15","83","211","255","143","35","3"],["1","17","111","353","579","489","207","41","3"],["1","19","143","547","1143","1323","839","283","47","3"],["1","21","179","801","2043","3045","2651","1329","371","53","3"],["1","23","219","1123","3391","6231","7019","4819","1983","471","59","3"],["1","25","263","1521","5315","11665","16295","14489","8131","2825","583","65","3"],["1","27","311","2003","7959","20371","34191","37803","27439","12939","3879","707","71","3"],["1","29","363","2577","11483","33645","66227","88289","79731","48509","19643","5169","843","77","3"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","9","22","53","128","309","746","1801","4348","10497","25342","61181","147704","356589"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A102662","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","3"]],"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":159009,"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-5b6052467d550b8d"]},{"id":159010,"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-5b6052467d550b8d"]},{"id":159011,"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-5b6052467d550b8d"]},{"id":159012,"position":3,"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-5b6052467d550b8d"]},{"id":159013,"position":4,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-1 - sqrt(2), -1 + sqrt(2)]","output_interval":"(-inf, +inf)","statement":"every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] has a real-rooted image"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in [-1 - sqrt(2), -1 + sqrt(2)] to roots in (-inf, +inf): every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] has a real-rooted image","region_exact":null,"edges":["e-28cf6d258736730b"]},{"id":159014,"position":5,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-1 - sqrt(2), -1 + sqrt(2)]","output_interval":"[-3 - 2 sqrt(2), -3 + 2 sqrt(2)]","statement":"every real polynomial with all roots in E = [-1 - sqrt(2), -1 + 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 [-1 - sqrt(2), -1 + sqrt(2)] to roots in [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]: every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] has an image with all roots in I = [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]","region_exact":null,"edges":["e-28cf6d258736730b"]},{"id":159015,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"[-1 - sqrt(2), 0]","output_interval":"[-3 - 2 sqrt(2), -3 + 2 sqrt(2)]","statement":"every PF input with all roots in [-1 - 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 [-1 - sqrt(2), 0] to roots in [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]: every PF input with all roots in [-1 - sqrt(2), 0] has a PF image; no larger [-R, 0] works","region_exact":null,"edges":["e-28cf6d258736730b"]}],"region_exact":{},"incoming_edges":[{"id":"e-28cf6d258736730b","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-14dfd91463cae288","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":"108f74e95a83f20278ebcfb6318ab6e8a446475ffdadfc3989408f30f55434e8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"[-1 - sqrt(2), -1 + sqrt(2)]","output_interval":"(-inf, +inf)","statement":"every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] has a real-rooted image"},{"claim":"basis_map_preserver","input_interval":"[-1 - sqrt(2), -1 + sqrt(2)]","output_interval":"[-3 - 2 sqrt(2), -3 + 2 sqrt(2)]","statement":"every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] has an image with all roots in I = [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]"},{"claim":"basis_map_preserver","input_interval":"[-1 - sqrt(2), 0]","output_interval":"[-3 - 2 sqrt(2), -3 + 2 sqrt(2)]","statement":"every PF input with all roots in [-1 - sqrt(2), 0] has a PF image; no larger [-R, 0] works"}],"premises":[]},{"id":"e-5b6052467d550b8d","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-14dfd91463cae288","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"sign","lean_declarations":["RealRooted.threeTerm_interlaces_of_nonnegCoeffs","RealRooted.LiuWang.strictInterl_nonpos_lag_sequence"],"certificate_hash":"bf1d5416c07186ddddd49dc2f4720a04cfe1666748bec3181fa6853a08055124","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-06cbef974e179f0a","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-23e2b6ed6bdac95d","conclusion_name":"basis_map_A013610 of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-30bb9f6b8a15c298","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-3dddab772a6867d2","conclusion_name":"basis_map_A013609 of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-38bf0b24f025cc0b","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-baf4dd53b457b011","conclusion_name":"basis_map_A013611 of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-8d8fb2debc89383b","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-6154960a0891a0b7","conclusion_name":"basis_map_A007318 of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-ac324c30ef49a129","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-96f82186178adef0","conclusion_name":"basis_map_A008277 of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-0e2d2744a82d78ea","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-df439a8e24e49dda","conclusion_name":"one_plus_derivative of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-1ea4e83a9ca9caed","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-1ff351802844a488","conclusion_name":"mobius_x_over_x_plus_one of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-21b50016cc9ebe72","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-267d61095455b8c8","conclusion_name":"reversal of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-2f65580ab8dffd6d","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-310dbc4638e545d5","conclusion_name":"derivative of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-3e98f257bdd1a607","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8a12c68ee675183d","conclusion_name":"theta_plus_one of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-6f86655597309e29","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-6154960a0891a0b7","conclusion_name":"basis_map_A007318 of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-eaf3df36dd50a693","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-1ab9c1cd3df94b60","conclusion_name":"theta_plus_one_squared of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-433ff3d90f25f7b2","operation":"veronese_section","operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-583675232a075e94","conclusion_name":"veronese_section_2_0 of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."},{"id":"e-d84f036c2dc882d3","operation":"veronese_section","operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-f7ad1fd21f0cb4ed","conclusion_name":"veronese_section_2_1 of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)."}],"used_by_total":14,"links":[{"id":"l-a9cc9a679e1915ea","kind":"fingerprint_related","node_a_id":"n-14dfd91463cae288","node_b_id":"n-267d61095455b8c8","fingerprint":"f6-867f73b103ecf726","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-867f73b103ecf726"},"other_id":"n-267d61095455b8c8","other":{"id":"n-267d61095455b8c8","name":"reversal of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).","first_index":0,"fingerprint":"f6-867f73b103ecf726","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["3","5","1"]],"card_rows":[["1"],["3","1"],["3","5","1"],["3","11","7","1"]],"row_sums_preview":["1","4","9","22","53","128","309","746"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-14dfd91463cae288","steps":["reverse"],"edges":["e-21b50016cc9ebe72"],"text":"A102662 \u2192 reverse"},"headline":"A102662 \u2192 reverse","chain":null}},{"id":"l-40b679f67b4780f2","kind":"oeis_match","node_a_id":"n-14dfd91463cae288","node_b_id":null,"fingerprint":null,"oeis_id":"A102662","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"triangle","terms_compared":69},"in_catalog":true,"oeis_id":"A102662","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-63b8bb07730ab0b1","kind":"oeis_match","node_a_id":"n-14dfd91463cae288","node_b_id":null,"fingerprint":null,"oeis_id":"A048654","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":"A048654","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A102662","relation":"identification"},{"oeis_id":"A048654","relation":"match_row_sums"},{"oeis_id":"A102662","relation":"match_triangle"}],"bfiles":[{"node_id":"n-14dfd91463cae288","oeis_id":"A048654","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":26,"oeis_terms_available":26,"term_count":200,"raw_size":8646},{"node_id":"n-14dfd91463cae288","oeis_id":"A102662","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":69,"oeis_terms_agreeing":69,"oeis_terms_available":69,"term_count":990,"raw_size":12366}],"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,  3;\n  2: 1,  5,   3;\n  3: 1,  7,  11,   3;\n  4: 1,  9,  23,  17,    3;\n  5: 1, 11,  39,  51,   23,    3;\n  6: 1, 13,  59, 113,   91,   29,   3;\n  7: 1, 15,  83, 211,  255,  143,  35,   3;\n  8: 1, 17, 111, 353,  579,  489, 207,  41,  3;\n  9: 1, 19, 143, 547, 1143, 1323, 839, 283, 47, 3;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*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] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*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}},{"oeis_id":"A102662","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  3;\n   3: 1,  5,   3;\n   4: 1,  7,  11,   3;\n   5: 1,  9,  23,  17,    3;\n   6: 1, 11,  39,  51,   23,    3;\n   7: 1, 13,  59, 113,   91,   29,   3;\n   8: 1, 15,  83, 211,  255,  143,  35,   3;\n   9: 1, 17, 111, 353,  579,  489, 207,  41,  3;\n  10: 1, 19, 143, 547, 1143, 1323, 839, 283, 47, 3;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + x*P[n - 2, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = (1 + x)*v[i - 1] + x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-14dfd91463cae288","steps":[],"edges":[],"text":"A102662"},"headline":"A102662: Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).","chain":null}
