{"id":"n-89d60963344cadbc","revision":1,"first_index":0,"fingerprint":"f6-5fc9190b1f03dd89","name":"basis_map_A013610 of Triangular array formed by the Mersenne numbers.","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","3"],["14","24","9"],["48","132","108","27"],["164","624","810","432","81"],["560","2724","4896","4104","1620","243"],["1912","11328","26136","30240","18630","5832","729"],["6528","45600","128736","191160","162000","78732","20412","2187"],["22288","179328","599472","1090368","1184220","789264","316386","69984","6561"],["76096","692976","2678400","5777568","7683984","6552252","3592512","1224720","236196","19683"],["259808","2641536","11593584","28964736","45700200","47682432","33394032","15536448","4605822","787320","59049"],["887040","9959616","48942144","139084560","254327040","314725824","269438400","159878448","64560240","16927380","2598156","177147"],["3028544","37216512","202460256","645235200","1343161440","1926519552","1955143008","1416756096","728664660","259815600","61056666","8503056","531441"],["10340096","138032448","823606272","2910152448","6799697280","11106110880","13061253888","11212696512","7035806448","3191401620","1018477152","216827928","27634932","1594323"],["35303296","508717056","3303601920","12820958208","33242923584","60970496256","81653062176","81201035520","60323986008","33353394624","13526000136","3905737056","759960630","89282088","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","47","315","2111","14147","94807","635355","4257871","28534387","191224967","1281505995","8588092031","57553632227","385699241527"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167174,"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-5de408358bde06b7"]},{"id":167175,"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-5de408358bde06b7"]},{"id":167176,"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-5de408358bde06b7"]},{"id":167177,"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-5de408358bde06b7"]}],"region_exact":{},"incoming_edges":[{"id":"e-5de408358bde06b7","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-89d60963344cadbc","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":"0f6ae050b83111ca11c5124068e5a521f8c130eaf533e39ba16b05bc4caf5f74","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-9b5c7e97e5191da6","role":"sequence","name":"Triangular array formed by the Mersenne numbers.","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-b1b8927358c66c57","kind":"oeis_match","node_a_id":"n-89d60963344cadbc","node_b_id":null,"fingerprint":null,"oeis_id":"A186446","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":"A186446","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A186446","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-89d60963344cadbc","oeis_id":"A186446","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":17433}],"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,      3;\n  2:    14,     24,       9;\n  3:    48,    132,     108,      27;\n  4:   164,    624,     810,     432,      81;\n  5:   560,   2724,    4896,    4104,    1620,     243;\n  6:  1912,  11328,   26136,   30240,   18630,    5832,     729;\n  7:  6528,  45600,  128736,  191160,  162000,   78732,   20412,    2187;\n  8: 22288, 179328,  599472, 1090368, 1184220,  789264,  316386,   69984,   6561;\n  9: 76096, 692976, 2678400, 5777568, 7683984, 6552252, 3592512, 1224720, 236196, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -2*P(n-2,x) + (4+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-2*P[n - 2, x] + (4 + 3*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -2*v[i - 2] + (4 + 3*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-9b5c7e97e5191da6","steps":["basis A013610"],"edges":["e-5de408358bde06b7"],"text":"A110441 \u2192 basis A013610"},"headline":"A110441 \u2192 basis A013610","chain":null}
