{"id":"n-ff69dadb0811f5e6","revision":1,"first_index":0,"fingerprint":"f6-77231177ad534f16","name":"Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).","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"],["9","7","1"],["27","38","12","1"],["81","192","101","18","1"],["243","969","755","215","25","1"],["729","5115","5494","2205","400","33","1"],["2187","29322","40971","21469","5355","679","42","1"],["6561","187992","323658","209356","66619","11452","1078","52","1"],["19683","1370745","2764926","2111318","813645","176295","22302","1626","63","1"],["59049","11392839","25811091","22448130","10076345","2619981","414813","40410","2355","75","1"],["177147","107043606","263989242","254297241","129031265","38847248","7343490","891363","69135","3300","88","1"],["531441","1122823944","2951126991","3083362326","1728215566","586433364","127967279","18489042","1781538","112860","4499","102","1"],["1594323","12989320785","35886116097","40046861205","24361470276","9126215098","2241576909","373213841","42723252","3355638","177177","5993","117","1"],["4782969","164040593067","472073225688","556384764663","362429978547","147510010648","40000721761","7491925727","987305891","92011920","6015009","269087","7826","133","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","17","78","393","2208","13977","100026","806769","7280604","72865089","801693126","9620848953","125072630712","1751021612937"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A327997","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["3"],["-3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159914,"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-ea394865db18a818"]},{"id":159915,"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-a526bc158f95a9e3","e-ea394865db18a818"]},{"id":159916,"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-a526bc158f95a9e3"]},{"id":159917,"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-a526bc158f95a9e3"]},{"id":159918,"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-a526bc158f95a9e3"]}],"region_exact":{},"incoming_edges":[{"id":"e-a526bc158f95a9e3","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-ff69dadb0811f5e6","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":"9b55553c1ca89452e512ff125a94bdbe438a3d2e9f523b6e9f6102352d2ddaf4","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":[]},{"id":"e-ea394865db18a818","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-ff69dadb0811f5e6","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":"c730eae8e0ec70e4863e53cc9878f9393629e2e85b2cab5460c006fa5465e85c","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":[]}],"used_by":[{"id":"e-5617fd17362674c9","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-3a6170eba9ce35d2","conclusion_name":"basis_map_A013610 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3)."},{"id":"e-8d805b4c99a48239","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-ad08ac6743aea7a3","conclusion_name":"basis_map_A008277 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3)."},{"id":"e-9953a8fe10a45b2a","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-64adfdb4384854aa","conclusion_name":"basis_map_A013609 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3)."},{"id":"e-cae95ae48665cd8b","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-e0a0f9191d8a0512","conclusion_name":"basis_map_A007318 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3)."},{"id":"e-e65e9efc7834c6f3","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-957f8e2516358b82","conclusion_name":"basis_map_A013611 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3)."}],"used_by_total":5,"links":[{"id":"l-2388214778d106bd","kind":"oeis_match","node_a_id":"n-ff69dadb0811f5e6","node_b_id":null,"fingerprint":null,"oeis_id":"A053486","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":"A053486","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e2d681173d4f46f6","kind":"oeis_match","node_a_id":"n-ff69dadb0811f5e6","node_b_id":null,"fingerprint":null,"oeis_id":"A327997","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":"A327997","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A327997","relation":"identification"},{"oeis_id":"A053486","relation":"match_row_sums"},{"oeis_id":"A327997","relation":"match_triangle"}],"bfiles":[{"node_id":"n-ff69dadb0811f5e6","oeis_id":"A053486","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":34322},{"node_id":"n-ff69dadb0811f5e6","oeis_id":"A327997","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":26529}],"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:     9,       7,       1;\n  3:    27,      38,      12,       1;\n  4:    81,     192,     101,      18,      1;\n  5:   243,     969,     755,     215,     25,      1;\n  6:   729,    5115,    5494,    2205,    400,     33,     1;\n  7:  2187,   29322,   40971,   21469,   5355,    679,    42,    1;\n  8:  6561,  187992,  323658,  209356,  66619,  11452,  1078,   52,  1;\n  9: 19683, 1370745, 2764926, 2111318, 813645, 176295, 22302, 1626, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+2+x)*P(n-1,x) - (3*n-3)*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[(n + 2 + x)*P[n - 1, x] - (3*n - 3)*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] = (n + 2 + x)*v[i - 1] - (3*n - 3)*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":"A327997","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:     9,       7,       1;\n  3:    27,      38,      12,       1;\n  4:    81,     192,     101,      18,      1;\n  5:   243,     969,     755,     215,     25,      1;\n  6:   729,    5115,    5494,    2205,    400,     33,     1;\n  7:  2187,   29322,   40971,   21469,   5355,    679,    42,    1;\n  8:  6561,  187992,  323658,  209356,  66619,  11452,  1078,   52,  1;\n  9: 19683, 1370745, 2764926, 2111318, 813645, 176295, 22302, 1626, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+2+x)*P(n-1,x) - (3*n-3)*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[(n + 2 + x)*P[n - 1, x] - (3*n - 3)*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] = (n + 2 + x)*v[i - 1] - (3*n - 3)*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-ff69dadb0811f5e6","steps":[],"edges":[],"text":"A327997"},"headline":"A327997: Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).","chain":null}
