{"id":"n-096818d27dad5765","revision":1,"first_index":0,"fingerprint":"f6-4349cc41c74631b0","name":"basis_map_A013609 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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"],["4","6"],["10","24","12"],["28","96","96","24"],["76","336","504","288","48"],["208","1128","2256","2016","768","96"],["568","3648","9120","11136","6816","1920","192"],["1552","11520","34560","53568","45504","20736","4608","384"],["4240","35712","124992","235008","256320","163584","58752","10752","768"],["11584","109152","436608","965376","1287936","1063296","536832","158208","24576","1536"],["31648","329856","1484352","3773952","5959296","6054912","3972096","1646592","410112","55296","3072"],["86464","987648","4938240","14193792","25903872","31306752","25380864","13701120","4795392","1032192","122880","6144"],["236224","2934528","16139904","51749376","107209728","150368256","145539072","97345536","44399616","13406208","2537472","270336","12288"],["645376","8662656","51975936","183919104","426501120","681384960","767803392","614694912","347885568","136857600","36261888","6119424","589824","24576"],["1763200","25430016","165295104","639848448","1642257408","2945347584","3790927872","3541843968","2403342336","1173897216","404938752","95453184","14524416","1277952","49152"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","46","244","1252","6472","33400","172432","890128","4595104","23721184","122455360","632148544","3263326336","16846196608"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166438,"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":true,"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-e4b64dd489e5e76c"]},{"id":166439,"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-e4b64dd489e5e76c"]},{"id":166440,"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-e4b64dd489e5e76c"]},{"id":166441,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-e4b64dd489e5e76c"]}],"region_exact":{},"incoming_edges":[{"id":"e-e4b64dd489e5e76c","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-096818d27dad5765","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"60b9921335533c520bb9aa1218ce3335644579f369e3f638d51e6fecaf8994fb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-eac80c2bf8e566f5","role":"sequence","name":"Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-a9bb07783525903a","kind":"oeis_match","node_a_id":"n-096818d27dad5765","node_b_id":null,"fingerprint":null,"oeis_id":"A138041","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":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A138041","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A138041","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-096818d27dad5765","oeis_id":"A138041","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":15228}],"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:     4,      6;\n  2:    10,     24,     12;\n  3:    28,     96,     96,     24;\n  4:    76,    336,    504,    288,      48;\n  5:   208,   1128,   2256,   2016,     768,      96;\n  6:   568,   3648,   9120,  11136,    6816,    1920,    192;\n  7:  1552,  11520,  34560,  53568,   45504,   20736,   4608,    384;\n  8:  4240,  35712, 124992, 235008,  256320,  163584,  58752,  10752,   768;\n  9: 11584, 109152, 436608, 965376, 1287936, 1063296, 536832, 158208, 24576, 1536;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+4*x)*P(n-2,x) + (2+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 6*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 4*x)*P[n - 2, x] + (2 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 4*x)*v[i - 2] + (2 + 2*x)*v[i - 1]); 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-eac80c2bf8e566f5","steps":["basis A013609"],"edges":["e-e4b64dd489e5e76c"],"text":"A249250 \u2192 basis A013609"},"headline":"A249250 \u2192 basis A013609","chain":null}